#4320264

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1922 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)(3,8)(4,5)(6,7)(9,10,14,13)(11,16,15,12)(19,23)(22,24)(25,26,29,32)(27,30,28,31);(2,7)(3,5)(9,10,14,13)(11,12,15,16)(18,21)(22,24);(1,3,6,5)(2,8,7,4)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,4,6,8)(2,3,7,5)(9,12,14,16)(10,15,13,11)(25,27,29,28)(26,30,32,31);(1,5,6,3)(2,8,7,4)(9,13,14,10)(11,16,15,12)(17,19)(18,22)(20,23)(21,24)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4227460);