#712738

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

2022 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=i,d^2=fg,e^2=f,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, 619934);