#4375535

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2808 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)(6,9)(10,12)(15,19)(18,20)(21,22,27,26)(23,30,24,28)(25,33,34,29)(31,35,32,36);(14,17)(18,20)(21,23,27,24)(22,28,26,30)(25,31,34,32)(29,35,33,36);(1,2)(3,4)(13,14,16,17)(15,18,19,20)(21,24,27,23)(22,28,26,30)(25,31,34,32)(29,36,33,35);(5,6,8,9)(7,10,11,12)(21,25,27,34)(22,29,26,33)(23,31,24,32)(28,35,30,36);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,19)(17,20)(21,26,27,22)(23,28,24,30)(25,33,34,29)(31,35,32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,d^2=i,e^2=i,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, 4282731);