#7601696

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1321 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,d^2=gi,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=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, 7508892);