#5493242

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2238 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=hi,e^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	#72267557
maximal normal subgroup of	#72267557
maximal quotient of	#72267561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5400438);