#2007418

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fghi,d^2=fghi,e^2=fhi,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, 1914614);