#6443124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5092 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=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, 6350320);