#1776111

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18,21,22)(19,23,24,20)(26,30)(28,35)(31,36)(34,39);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(17,19,21,24)(18,20,22,23)(26,30)(27,33)(32,37)(34,39);(3,7)(5,8)(11,16)(12,14)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1683307);