#6051929

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

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

2112 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,23)(21,22)(25,26,30,29)(27,32,31,28);(9,11,13,16)(10,14,15,12)(18,21)(23,24)(26,29)(27,31);(2,6)(3,8)(10,15)(11,16)(17,19,22,20)(18,23,21,24)(25,27)(26,28)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20,22,19)(18,24,21,23)(25,28,30,32)(26,31,29,27);(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,18)(19,24,20,23)(25,29,30,26)(27,32,31,28)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,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, 5959125);