#7065229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972425);