#6685642

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1849 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,33)(24,32)(25,34)(26,31)(27,37)(28,35)(29,38)(30,36)(39,43)(40,42)(41,44);(6,9)(7,11)(13,15,17,25)(14,19,20,34)(16,27,28,24)(18,23,30,26)(21,37,35,32)(22,33,36,31)(29,40,41,39)(38,42,44,43);(1,2,3,4)(14,20)(15,23)(16,28)(18,30)(19,31)(24,40)(25,26)(27,39)(32,43)(33,34)(37,42)(38,44);(6,9)(10,12)(13,16,17,28)(14,21,20,35)(15,24,25,27)(18,29,30,41)(19,32,34,37)(22,38,36,44)(23,39,26,40)(31,42,33,43);(5,7,8,11)(6,10,9,12)(15,23)(18,30)(19,33)(22,36)(24,39)(25,26)(27,40)(29,41)(31,34)(32,43)(37,42)(38,44)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,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, 6592838);