#5958782

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

3602 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)(9,10,12,13)(11,15,16,14)(17,18,21,22)(19,24,23,20)(25,26,29,31)(27,32,28,30);(2,5)(6,8)(10,13)(11,16)(25,27,29,28)(26,30,31,32);(11,16)(14,15)(18,22)(19,23)(27,28)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,14,13,15)(17,19,21,23)(18,20,22,24)(25,28,29,27)(26,30,31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,22)(21,24)(25,26)(27,32)(28,30)(29,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,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	#72336306, #72336404, #72336437
maximal normal subgroup of	#72336306, #72336404, #72336437

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5865978);