#6443257

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350453);