#5989660

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	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,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, 5896856);