#1538068

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1879 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:

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

Polycyclic presentation:

2^9|b^2=fgi,c^2=fhi,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1445264);