Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,29,38)(20,24,32,41)(21,25,35,44)(27,36,45,47)(28,37,30,39)(31,40,46,48)(33,42,34,43);(2,5)(6,8)(17,19)(18,23)(20,31)(21,28)(22,29)(24,40)(25,37)(26,38)(27,33)(30,35)(32,46)(34,45)(36,42)(39,44)(41,48)(43,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,24)(19,27)(21,34)(22,32)(23,36)(25,43)(26,41)(28,31)(29,45)(30,46)(33,35)(37,40)(38,47)(39,48)(42,44);(9,11,12,15)(10,14,13,16)(19,28)(21,35)(23,37)(25,44)(27,31)(29,30)(33,34)(36,40)(38,39)(42,43)(45,46)(47,48);(19,29)(20,32)(23,38)(24,41)(28,30)(33,34)(37,39)(42,43)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356607);