#1287180

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=ghi,e^2=fg,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, 1194376);