#4320261

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2173 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,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, 4227457);