#7148803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7055999);