#7615939

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1463 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,21,26,31)(20,27,30,24)(25,32,28,29);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,22,26)(18,21,23,31)(20,28,30,25)(24,32,27,29);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,24)(19,25)(21,29)(22,30)(23,27)(26,28)(31,32);(10,14)(13,15)(17,21,23,26,22,31,18,19)(20,29,27,28,30,32,24,25);(2,5)(6,8)(10,14)(13,15)(20,30)(24,27)(25,28)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7523135);