#3537206

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=hi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3444402);