#6051893

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1896 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fghi,d^2=f,e^2=f,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, 5959089);