#6132628

order	512
nilpotency class	2
isoclinism	isoclinic class of #6131635
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1790 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6039824);