#2377507

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

2566 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2284703);