#6671152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1794 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578348);