#72312447

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄²
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	C₂×#1889
prime radical	C₂×#3025	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

4104 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=egj,e^2=j,g^2=j,h^2=j,c^b=cj,d^a=df,d^b=dh,e^d=ej,f^b=fi,g^d=gj,h^a=hij,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312460, #72312638, #72312952, …
maximal normal subgroup of	#72312460, #72312638, #72312952, …

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, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.7*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
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);