#6465642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6372838);