#1281869

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1432 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)(19,23)(20,21)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(10,13)(14,16)(17,18,22,24)(19,20,23,21)(26,31)(27,33)(34,39)(35,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,4,6,2)(3,8,7,5)(17,20,22,21)(18,23,24,19)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(1,5,6,8)(2,7,4,3)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,19,24,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)

Polycyclic presentation:

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

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, 1189065);