#6814789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,28)(17,19)(18,32)(20,27)(21,29)(22,25)(24,38)(26,41)(30,37)(31,36)(33,39)(34,44)(35,42)(40,43);(2,3)(5,7)(6,9)(8,12)(10,11)(14,19)(15,24)(16,29)(22,32)(27,39)(31,40)(34,35)(37,41);(1,2,4,3)(13,15,25,39,17,24,18,27)(14,20,32,38,19,33,22,23)(16,30,40,42,29,26,31,44)(21,35,43,37,28,34,36,41);(6,10)(9,11)(13,16)(14,21)(15,26)(17,29)(18,31)(19,28)(20,34)(22,36)(23,37)(24,30)(25,40)(27,42)(32,43)(33,35)(38,41)(39,44);(1,3,4,2)(7,12)(9,11)(15,24)(16,29)(20,33)(21,28)(23,38)(27,39)(31,40)(36,43)

Polycyclic presentation:

2^9|c^2=gh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721985);