#7615399

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gi,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7522595);