#5783928

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

1458 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=i,d^2=g,e^2=f,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, 5691124);