#6019448

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 2097152
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2050 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)(9,10,12,15)(11,14,16,13)(18,21)(22,23)(26,28)(27,32);(2,5)(6,8)(9,11,12,16)(10,13,15,14)(17,18,20,21)(19,23,24,22)(25,26,30,28)(27,29,32,31);(1,3,4,7)(2,6,5,8)(9,10,12,15)(11,14,16,13)(17,18)(19,22)(20,21)(23,24)(25,27,30,32)(26,31,28,29);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,20,24)(18,22,21,23)(25,28,30,26)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=fhi,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, 5926644);