#6668473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2661 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6575669);