#7205622

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7112818);