#6816402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(13,14,18,22)(15,23,26,37)(16,20,28,34)(17,30,24,21)(19,31,33,27)(25,36,40,44)(29,41,38,35)(32,42,43,39);(2,3)(5,7,8,12)(6,9,10,11);(1,2,4,3)(6,10)(7,12)(13,15,24,31,18,26,17,27)(14,19,30,37,22,33,21,23)(16,25,38,42,28,40,29,39)(20,32,41,44,34,43,35,36);(6,10)(9,11)(13,16)(14,20)(15,25)(17,29)(18,28)(19,32)(21,35)(22,34)(23,36)(24,38)(26,40)(27,39)(30,41)(31,42)(33,43)(37,44);(1,3,4,2)(15,26)(16,28)(19,33)(20,34)(23,37)(27,31)(29,38)(35,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6723598);