Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,7,8,6)(11,15)(13,16)(19,25)(20,27)(22,28)(23,30)(33,34,38,39)(35,40,41,36)(37,45,46,42)(43,44,48,47);(9,10,12,14)(11,16,15,13)(19,25)(22,28)(26,31)(29,32)(34,39)(36,40)(37,46)(43,48);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,45)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,3,4,8)(2,6,5,7)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926440);