#1377483

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1766 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1284679);