#4746690

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

2986 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)(6,8)(9,10,13,14)(11,12,16,15)(17,18,21,23)(19,20,24,22)(25,26,28,30)(27,31,32,29);(10,14)(11,16)(17,19)(18,22)(20,23)(21,24)(25,26,28,30)(27,29,32,31);(10,14)(12,15)(18,23)(19,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232685, #72232691, #72232704
maximal normal subgroup of	#72232685, #72232691, #72232704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4653886);