#1291408

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

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fhi,d^2=f,e^2=fh,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, 1198604);