#6685122

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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1869 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)(16,20)(17,22)(18,21)(23,32)(24,35)(25,33)(26,34)(27,36)(28,38)(29,37)(30,40)(31,39)(41,43)(42,44);(6,9)(7,11)(13,15,18,24)(14,19,21,35)(16,27,28,42)(17,25,31,26)(20,36,38,44)(22,33,39,34)(23,30,41,29)(32,40,43,37);(1,2,3,4)(13,16,18,28)(14,20,21,38)(15,23,24,41)(17,30,31,29)(19,32,35,43)(22,40,39,37)(25,27,26,42)(33,36,34,44);(6,9)(10,12)(14,21)(15,24)(16,28)(22,39)(25,26)(29,30)(32,43)(36,44);(5,7,8,11)(6,10,9,12)(15,25)(17,31)(19,33)(22,39)(23,27)(24,26)(29,30)(32,36)(34,35)(37,40)(41,42)(43,44)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,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, 6592318);