#4112031

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

1444 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=ghi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4019227);