#3501294

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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2212 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=ghi,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, 3408490);