#6342415

order	512
nilpotency class	2
isoclinism	isoclinic class of #6342371
automorphism	Aut(G) order 15728640
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2361 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:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=h,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249611);