#4364137

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

2240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,e^2=fi,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, 4271333);