#1412360

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
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

1530 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fh,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,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, 1319556);