#720660

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2072 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)(7,11)(10,12)(15,19)(18,20)(21,22)(23,28)(24,25)(26,27)(29,34)(30,35)(31,36)(32,33);(6,9)(10,12)(13,14,17,16)(15,20,19,18)(22,27)(23,30)(25,29)(31,33);(5,6,8,9)(7,10,11,12)(13,15,17,19)(14,18,16,20)(21,23,26,30)(22,28,27,35)(24,33,34,31)(25,32,29,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,24)(22,29)(23,31)(25,27)(26,34)(28,32)(30,33)(35,36);(21,25,26,29)(22,24,27,34)(23,32,30,36)(28,33,35,31)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,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, 627856);