#2089240

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

1878 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)(7,8)(9,10)(11,12)(14,15)(16,20)(21,22,26,28)(23,29,30,24)(25,33,34,27)(31,32,36,35);(5,6,9,10)(7,8,11,12)(13,14,17,15)(16,18,20,19)(23,30)(24,29)(31,36)(32,35);(6,10)(8,12)(13,15,17,14)(16,18,20,19)(21,23)(22,24)(25,31)(26,30)(27,32)(28,29)(33,35)(34,36);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8)(6,11)(7,10)(9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,34)(22,27,28,33)(23,31,30,36)(24,32,29,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,d^2=fgh,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, 1996436);