#5482519

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(19,23)(21,24)(26,30)(27,33)(32,37)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18,20,22)(19,21,23,24)(28,36)(32,37)(35,39)(38,40);(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,12,16,15)(17,18)(19,21)(20,22)(23,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,23)(22,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303954, #72303956, #72304090
maximal normal subgroup of	#72303954, #72303956, #72304090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5389715);