#7205609

order	512
nilpotency class	3
isoclinism	isoclinic class of #7199982
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 ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,e^2=fi,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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7112805);