#720291

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2157 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)(6,10)(7,12)(14,18)(19,20)(21,22,26,30)(23,31,34,27)(24,25,29,28)(32,33,35,36);(14,18)(19,20)(21,23,26,34)(22,27,30,31)(24,32,29,35)(25,36,28,33);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,35)(25,33)(26,34)(28,36)(29,32)(30,31);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,15,17,16)(14,19,18,20)(21,24,26,29)(22,28,30,25)(23,32,34,35)(27,33,31,36);(5,8,9,11)(6,12,10,7)(13,16,17,15)(14,20,18,19)(21,25,26,28)(22,29,30,24)(23,33,34,36)(27,32,31,35)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=fhi,e^2=g,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, 627487);