#6408486

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	#2424	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,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, 6315682);