#5331570

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1532 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26,30,33)(27,29,34,31)(28,32,38,40)(35,37,36,39);(4,8)(6,7)(9,10,12,13)(11,15,16,14)(19,23)(21,24)(27,34)(29,31)(35,36)(37,39);(2,3)(6,7)(10,13)(14,15)(17,18,20,22)(19,21,23,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,37)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,23)(22,24)(25,29,30,31)(26,27,33,34)(28,37,38,39)(32,35,40,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=ghi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5238766);