#1142706

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1894 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:

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

Polycyclic presentation:

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

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, 1049902);