#1858676

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

1360 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=gi,d^2=fghi,e^2=fgh,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, 1765872);