#1115231

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

1930 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=fg,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, 1022427);