#6628956

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4572 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)(5,6)(7,8)(9,10)(11,20)(12,18)(13,19)(14,16)(15,27)(17,22)(21,35)(23,32)(24,33)(25,30)(26,34)(28,29)(31,38)(36,39)(37,40);(1,2,3,4)(10,16)(17,29)(18,30)(19,27)(20,32)(31,39)(33,34)(35,40);(9,11,14,23)(10,17,16,29)(12,21,25,37)(13,22,15,28)(18,31,30,39)(19,32,27,20)(24,36,26,38)(33,35,34,40);(6,8)(9,12)(10,18)(11,21)(13,26)(14,25)(15,24)(16,30)(17,31)(19,34)(20,35)(22,38)(23,37)(27,33)(28,36)(29,39)(32,40);(9,13,14,15)(10,19,16,27)(11,22,23,28)(12,24,25,26)(17,32,29,20)(18,33,30,34)(21,36,37,38)(31,35,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536152);