#6629992

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3170 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)(6,8)(9,10)(11,16)(12,24)(13,18)(14,27)(15,19)(17,25)(20,29)(21,35)(22,31)(23,37)(26,32)(28,33)(30,36)(34,40)(38,39);(1,2,3,4)(9,11)(10,16)(12,21)(13,22)(14,23)(15,20)(17,30)(18,31)(19,29)(24,35)(25,36)(26,34)(27,37)(28,38)(32,40)(33,39);(5,6,7,8)(11,20)(16,29)(21,34)(22,23)(30,39)(31,37)(35,40)(36,38);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,30)(18,32)(19,33)(20,34)(22,36)(23,38)(24,27)(29,39)(31,40)(35,37);(9,13,15,14)(10,18,19,27)(11,22,20,23)(12,25,26,28)(16,31,29,37)(17,32,33,24)(21,36,34,38)(30,40,39,35)

Polycyclic presentation:

2^9|b^2=f,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537188);