#2117098

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2024294);