#748895

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2487 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)(13,14,17,16)(15,19,20,18)(22,26)(23,29)(28,32)(31,35);(15,20)(18,19)(21,22,25,26)(23,27,29,30)(24,32,33,28)(31,36,35,34);(1,2,3,4)(13,15,17,20)(14,18,16,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,6)(7,10)(8,9)(11,12)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fg,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, 656091);