#4929690

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
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

2117 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)(9,10,13,14)(11,12,16,15)(17,18,22,20)(19,24,21,23)(25,26,29,30)(27,32,31,28);(2,3)(6,7)(10,14)(11,16)(17,19,22,21)(18,23,20,24)(25,27)(26,28)(29,31)(30,32);(10,14)(11,16)(18,20)(23,24)(26,30)(27,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,20,22,18)(19,23,21,24)(25,26)(27,32)(28,31)(29,30);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,16,14,11)(17,21,22,19)(18,23,20,24)(25,28,29,32)(26,31,30,27)

Polycyclic presentation:

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

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, 4836886);