#7601216

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1312 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)(12,14)(13,15)(17,18,23,20)(19,26,30,32)(21,25,22,31)(24,28,27,29);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,29,23,30,21,28)(18,24,25,26,20,27,31,32);(1,3,4,7)(2,5,6,8)(9,11,13,12)(10,14,15,16)(17,20,23,18)(19,27,30,24)(21,25,22,31)(26,29,32,28);(2,6)(5,8)(9,11,13,12)(10,14,15,16)(17,21,23,22)(18,25,20,31)(19,28,30,29)(24,26,27,32);(9,12,13,11)(10,14,15,16)(17,19,21,28,23,30,22,29)(18,24,31,32,20,27,25,26)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,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,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7508412);