#7612136

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 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)(11,13)(12,14)(15,16)(17,18,23,22)(19,24,27,32)(20,25,28,29)(21,30,31,26);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,27)(18,24,22,32)(20,26,28,30)(21,29,31,25);(10,14)(11,15)(17,20)(18,25)(19,26)(21,24)(22,29)(23,28)(27,30)(31,32);(10,14)(13,16)(17,21,22,30,23,31,18,26)(19,20,24,29,27,28,32,25);(1,4,5,3)(2,7,8,6)(20,28)(21,31)(25,29)(26,30)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

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, 7519332);