#726559

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=fgi,e^2=fg,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, 633755);