#6685202

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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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)(16,27)(17,20)(18,22)(21,29)(23,38)(24,36)(25,34)(26,32)(28,33)(30,41)(31,35)(37,42)(39,43)(40,44);(6,9)(7,11)(13,15,17,25)(14,19,20,34)(16,28,29,24)(18,23,31,26)(21,36,27,33)(22,38,35,32)(30,40,42,39)(37,43,41,44);(1,2,3,4)(14,20)(15,23)(16,29)(18,31)(19,32)(24,40)(25,26)(28,39)(33,44)(34,38)(36,43)(37,41);(6,9)(10,12)(13,16,17,29)(14,21,20,27)(15,24,25,28)(18,30,31,42)(19,33,34,36)(22,37,35,41)(23,39,26,40)(32,43,38,44);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(18,31)(19,32)(21,27)(24,39)(25,26)(28,40)(30,42)(33,43)(34,38)(36,44)

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=dfi,e^a=ei,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, 6592398);