#751615

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2314 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 658811);