#1175390

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1664 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=fi,d^2=fgh,e^2=fg,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, 1082586);