#7576366

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1357 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7483562);