#5323333

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
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) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fhi,d^2=hi,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, 5230529);