#5956763

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3657 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:

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

Polycyclic presentation:

2^9|a^2=f,d^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.

Relationships

maximal subgroup of	#72336285, #72336404, #72336449
maximal normal subgroup of	#72336285, #72336404, #72336449

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863959);