#1096951

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

1826 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)(5,6,10,8)(7,12,9,11)(13,14,17,16)(15,18,19,20)(22,26)(23,29)(24,32)(34,36);(15,19)(18,20)(21,22,25,26)(23,27,29,30)(24,33,32,28)(31,36,35,34);(5,7,10,9)(6,11,8,12)(13,15,17,19)(14,18,16,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2)(3,4)(5,8,10,6)(7,12,9,11)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,9,10,7)(6,12,8,11)(13,16,17,14)(15,20,19,18)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=gh,d^2=fg,e^2=fgh,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, 1004147);