#2563783

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2150 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=hi,d^2=f,e^2=hi,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, 2470979);