#6520048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6427244);