#5079949

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1618 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:

(1,2,6,7)(3,5,4,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,20)(19,24,21,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,39,38,40);(1,3,6,4)(2,5,7,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,20)(19,23,21,24)(26,30)(27,34)(28,37)(39,40);(2,7)(3,4)(10,14)(11,15)(17,19,22,21)(18,23,20,24)(25,27,29,34)(26,31,30,33)(28,38,37,35)(32,40,36,39);(1,4,6,3)(2,5,7,8)(9,11,13,15)(10,12,14,16)(17,20,22,18)(19,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40);(1,5)(2,3)(4,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,21,22,19)(18,23,20,24)(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=fghi,c^2=fgh,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, 4987145);