Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,38,41)(35,42,39,36)(37,45,48,40)(43,44,46,47);(10,13)(14,16)(19,26)(20,29)(22,25)(23,28)(33,35,38,39)(34,36,41,42)(37,46,48,43)(40,47,45,44);(9,10)(11,14)(12,13)(15,16)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31)(33,35)(34,36)(37,46)(38,39)(40,47)(41,42)(43,48)(44,45);(1,2)(3,6)(4,5)(7,8)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,36)(34,39)(35,41)(37,47)(38,42)(40,43)(44,48)(45,46);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37,38,48)(34,40,41,45)(35,43,39,46)(36,44,42,47)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4170363);