#6491475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398671);