#6467536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2293 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)(7,10)(8,9)(11,12)(13,14,17,22,18,23,30,19)(15,20,24,35,26,34,40,31)(16,21,27,37,29,38,41,32)(25,33,39,44,28,36,42,43);(2,4)(14,19)(15,24)(16,27)(17,30)(22,23)(25,28)(26,40)(29,41)(31,35)(32,37)(33,44)(36,43);(6,9)(10,12)(13,15,17,24,18,26,30,40)(14,20,22,35,23,34,19,31)(16,28,27,42,29,25,41,39)(21,36,37,43,38,33,32,44);(13,16,17,27,18,29,30,41)(14,21,22,37,23,38,19,32)(15,25,24,39,26,28,40,42)(20,33,35,44,34,36,31,43);(5,7,8,11)(6,10,9,12)(15,26)(20,34)(24,40)(25,28)(31,35)(33,36)(39,42)(43,44)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374732);