#6019446

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1803 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)(3,6)(4,5)(7,8)(9,10,13,14)(11,16,12,15)(17,18,20,22)(19,21,23,24)(25,26)(27,30)(28,29)(31,32);(2,5)(6,8)(10,14)(15,16)(19,23)(21,24)(25,27,29,32)(26,30,28,31);(3,7)(6,8)(9,11,13,12)(10,15,14,16)(17,18)(19,24)(20,22)(21,23)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,19,20,23)(18,21,22,24)(25,28,29,26)(27,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5926642);