#7283705

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234453
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×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1794 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

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, 7190901);