#4308694

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 6291456
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215890);