#7138662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#30574	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi

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, 7045858);