#5110638

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fhi,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5017834);