#1326438

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

1652 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=gi,d^2=i,e^2=fh,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, 1233634);