#1316557

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

1837 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 1223753);