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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(18,22)(19,25)(20,28)(30,32)(33,34,38,36)(35,41,43,39)(37,40,46,44)(42,48,45,47);(2,4)(3,8)(11,16)(13,15)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(3,8)(6,7)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=h,d^2=fh,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, 5908876);