#5992123

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2354 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)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,20)(19,23,24,21)(25,26,30,31)(27,32,29,28);(2,4)(3,8)(11,16)(13,15)(17,19,22,24)(18,23,20,21)(26,31)(27,29);(3,8)(6,7)(18,20)(19,24)(25,27,30,29)(26,32,31,28);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,23,24,21)(25,28,30,32)(26,27,31,29);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,21,22,23)(18,24,20,19)(25,29,30,27)(26,32,31,28)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=gh,d^2=fg,e^2=fg,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, 5899319);