#626315

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2326 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)(13,14,16,18)(15,19,20,17)(21,22)(23,30)(24,25)(26,34)(27,29)(28,35)(31,33)(32,36);(6,9)(10,12)(15,20)(17,19)(21,23,27,24)(22,25,29,30)(26,31,35,32)(28,33,34,36);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,17,18,19)(21,24,27,23)(22,25,29,30)(26,32,35,31)(28,33,34,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,23)(24,29)(26,33)(27,30)(28,31)(32,34)(35,36);(21,26,27,35)(22,28,29,34)(23,31,24,32)(25,33,30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 533511);