#6687016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1899 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:

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

Polycyclic presentation:

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

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, 6594212);