#1406278

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1871 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=f,e^2=fgh,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, 1313474);