#1316558

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

1838 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)(3,8)(4,7)(5,6)(9,10,13,15)(11,16,14,12)(17,18,20,22)(19,23,24,21);(1,2,6,5)(3,8,4,7)(9,11,13,14)(10,12,15,16)(25,26,28,30)(27,29,31,32);(1,3,6,4)(2,7,5,8)(9,11,13,14)(10,12,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,4,6,3)(2,7,5,8)(9,10)(11,12)(13,15)(14,16);(1,5,6,2)(3,8,4,7)(9,12)(10,14)(11,15)(13,16)(17,19,20,24)(18,21,22,23)(25,27,28,31)(26,29,30,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=f,d^2=fi,e^2=fgh,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, 1223754);