#72245189

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#246927
commutator, Frattini, cosocle	C₄×M₁₆	C₂⁴
LCS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=e,d^2=gi,f^2=hij,g^2=i,h^2=j,b^a=be,d^a=df,d^b=dj,d^c=dg,e^d=ei,f^a=fh,f^b=fij,f^c=fhi,f^e=fj,g^a=ghi,g^c=gi,h^c=hj,i^a=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245194, #72245425, #72306798, …
maximal normal subgroup of	#72245425, #72306798, #72306974

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