#720257

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

2242 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)(21,22,26,27)(23,28,29,35)(24,32,34,25)(30,31,33,36);(6,9)(10,12)(13,14,17,16)(15,20,19,18)(22,27)(23,29)(25,32)(30,33);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20)(21,23)(22,28)(24,33)(25,31)(26,29)(27,35)(30,34)(32,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,24)(22,25)(23,30)(26,34)(27,32)(28,36)(29,33)(31,35);(21,25)(22,24)(23,31)(26,32)(27,34)(28,33)(29,36)(30,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^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, 627453);