#6424568

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

1331 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)(17,18,22,20,21,24,31,30)(19,25,28,27,26,23,32,29);(3,8)(5,7)(9,10)(11,16)(12,15)(13,14)(19,26)(23,25)(27,29)(28,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,15,14,16)(17,19,21,26)(18,23,24,25)(20,29,30,27)(22,28,31,32);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,24,21,30,31,18)(19,27,28,23,26,29,32,25);(9,12,13,11)(10,15,14,16)(18,24)(19,26)(20,30)(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=dg,e^a=efi,c^b=cg,d^b=df,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, 6331764);