#4456933

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
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

1788 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(19,23)(22,24)(25,26)(27,28)(29,30)(31,32);(1,3,6,8)(2,7,4,5)(9,10,12,14)(11,13,16,15)(18,21)(22,24)(26,30)(28,31);(2,4)(3,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,31,30,28);(1,4,6,2)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(25,28,29,31)(26,27,30,32);(1,5,6,7)(2,8,4,3)(9,11,12,16)(10,13,14,15)(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4364129);