#2073142

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1520 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=fi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1980338);