#6408232

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	#34937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,20,25)(19,24,28,32,29,23,30,31);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,26)(25,32)(27,31);(10,14)(11,15)(17,19)(18,24)(20,30)(21,29)(22,28)(23,26)(25,31)(27,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,21,22)(18,25,26,27)(19,28,29,30)(23,31,24,32);(2,5)(6,8)(18,26)(19,29)(25,27)(28,30)

Polycyclic presentation:

2^9|a^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di,e^c=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, 6315428);