#7283611

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7190807);