#758235

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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

2157 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=h,d^2=i,e^2=i,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, 665431);