#1127555

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=h,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1034751);