#6667599

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2737 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)(13,14,19,18)(15,20,25,32)(16,21,27,33)(17,22,31,34)(23,35,39,26)(24,36,42,29)(28,37,30,38)(40,44,41,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,25)(14,20,18,32)(16,26,27,35)(17,29,31,36)(21,23,33,39)(22,24,34,42)(28,40,30,41)(37,44,38,43);(2,4)(13,16)(14,21)(15,23)(17,30)(18,33)(19,27)(20,35)(22,38)(24,41)(25,39)(26,32)(28,31)(29,44)(34,37)(36,43)(40,42);(6,10)(9,11)(16,27)(21,33)(23,39)(26,35)(28,30)(37,38)(40,41)(43,44);(7,12)(9,11)(13,17,19,31)(14,22,18,34)(15,24,25,42)(16,28,27,30)(20,36,32,29)(21,37,33,38)(23,40,39,41)(26,43,35,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6574795);