#1195947

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

1868 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)(5,6,9,12)(7,11,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(15,19)(17,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(5,7,9,8)(6,10,12,11)(13,15,16,19)(14,17,18,20)(21,24,26,23)(22,27,29,30)(25,31,34,32)(28,36,33,35);(1,2,3,4)(13,14)(15,17)(16,18)(19,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36);(5,8,9,7)(6,11,12,10)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fgh,d^2=fhi,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, 1103143);