#4310267

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(7,11)(10,12)(15,19)(18,20)(21,22,26,29)(23,30,24,27)(25,28,33,35)(31,36,32,34);(6,9)(10,12)(14,17)(18,20)(21,23,26,24)(22,27,29,30)(25,31,33,32)(28,34,35,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,24,26,23)(22,27,29,30)(25,32,33,31)(28,34,35,36);(13,15)(14,18)(16,19)(17,20)(21,22)(23,27)(24,30)(25,28)(26,29)(31,34)(32,36)(33,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,33)(22,28,29,35)(23,31,24,32)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4217463);