#6685608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(24,34)(25,35)(26,33)(27,36)(28,38)(29,37)(30,40)(31,39)(41,43)(42,44);(6,9)(7,11)(13,15,18,26)(14,19,21,33)(16,27,28,42)(17,24,31,25)(20,36,38,44)(22,34,39,35)(23,30,41,29)(32,40,43,37);(1,2,3,4)(13,16,18,28)(14,20,21,38)(15,23,26,41)(17,30,31,29)(19,32,33,43)(22,40,39,37)(24,27,25,42)(34,36,35,44);(6,9)(10,12)(14,21)(16,28)(19,33)(22,39)(23,41)(27,42)(29,30)(34,35);(5,7,8,11)(6,10,9,12)(15,24)(17,31)(19,34)(22,39)(23,27)(25,26)(29,30)(32,36)(33,35)(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,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, 6592804);