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 | |
1460 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)(3,7)(10,11)(12,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34)(35,36)(37,45)(38,39)(40,46)(41,44)(42,48)(43,47);(2,5)(6,8)(9,10,13,11)(12,15,16,14)(19,25)(23,26)(27,31)(30,32)(34,39)(35,41)(37,46)(43,48);(9,10,13,11)(12,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,38,41)(34,36,39,44)(37,42,46,47)(40,43,45,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=fh,e^2=fh,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, 5888292);