#5073799

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

1356 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,16,13,15)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,39,38,40);(2,6)(3,8)(9,10,14,12)(11,15,13,16)(17,19,21,18)(20,23,24,22)(26,30)(27,34)(28,37)(39,40);(3,8)(4,7)(9,11,14,13)(10,15,12,16)(25,27,29,34)(26,31,30,33)(28,38,37,35)(32,40,36,39);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,13,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,15,12,16)(25,28,29,37)(26,32,30,36)(27,35,34,38)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=fg,e^2=fgh,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, 4980995);