#1287240

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1903 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fi,d^2=h,e^2=f,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, 1194436);