#6424571

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18,23,27,22,24,21,26)(19,28,30,32,29,20,25,31);(3,8)(5,7)(9,10)(11,16)(12,15)(13,14)(18,24)(19,29)(25,30)(26,27);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,14,16)(17,19,22,29)(18,20,24,28)(21,25,23,30)(26,32,27,31);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,21,32,22,28,23,31)(18,25,26,29,24,30,27,19);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,26,24,27)(19,30,29,25)(20,31,28,32)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cg,d^b=dfi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6331767);