#1328840

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

1518 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=fg,d^2=g,e^2=fi,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, 1236036);