#5934127

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1714 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5841323);