#6424467

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	#2688	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(9,10)(11,13)(12,15)(14,16)(17,18,23,24,22,27,19,26)(20,29,30,32,31,25,28,21);(3,8)(5,7)(9,11,13,10)(12,15,16,14)(17,19,22,23)(18,24,27,26)(20,30,31,28)(21,25,32,29);(1,3,4,8)(2,5,6,7)(10,11)(14,15)(17,20)(18,25)(19,28)(21,24)(22,31)(23,30)(26,32)(27,29);(2,6)(5,7)(10,11)(14,15)(17,21,23,29,22,32,19,25)(18,20,24,30,27,31,26,28);(9,12,13,16)(10,14,11,15)(17,19,22,23)(18,26,27,24)(20,30,31,28)(21,29,32,25)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cg,d^b=dfi,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, 6331663);