#72312465

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24404
commutator	C₂×#41	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	C₂×#3008	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

2909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,9,7)(8,19,23,29)(10,15,26,18)(11,13,12,16)(14,28,17,25)(20,36,32,35)(21,31,24,34)(22,33,27,30);(5,7)(6,9)(8,20)(10,18)(11,16)(12,13)(14,21)(15,26)(17,24)(19,35)(22,30)(23,32)(25,31)(27,33)(28,34)(29,36);(1,3,4,2)(6,13)(7,16)(8,21)(10,26)(11,12)(14,31)(15,33)(17,34)(18,30)(19,32)(20,29)(23,24)(25,35)(28,36);(5,8,22,36)(6,14,30,34)(7,17,33,31)(9,23,27,35)(10,24,11,25)(12,28,26,21)(13,29,18,20)(15,32,16,19)

Polycyclic presentation:

2^10|a^2=e,c^2=ej,d^2=fhij,g^2=j,h^2=j,b^a=bj,c^a=cg,c^b=cg,d^a=dh,d^b=df,d^c=dfghj,f^a=fi,f^c=fij,f^d=fij,g^a=gj,g^b=gj,g^c=gj,h^a=hj,h^b=hij,h^c=hij,h^d=hi,h^f=hj,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312504, #72312638, #72312703
maximal normal subgroup of	#72312504, #72312638, #72312703

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5*F.10);;
gap> SetPower(coll, 4, F.6*F.8*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);