#1313162

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2190 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=g,d^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, 1220358);