Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1533 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)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,32)(21,36)(22,27)(23,26)(24,31)(25,35)(29,39)(30,40)(33,42)(34,43)(37,41)(38,44)(45,47)(46,48);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,22,31)(18,24,27,28)(20,33,35,46)(21,30,23,37)(25,42,32,48)(26,40,36,41)(29,38,47,34)(39,43,45,44);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,39)(26,43)(27,32)(28,45)(30,46)(31,47)(33,37)(36,44)(40,42)(41,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,37)(20,34,35,38)(24,40,28,41)(25,43,32,44)(29,46,47,33)(39,42,45,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(19,30)(20,35)(21,23)(24,41)(25,32)(26,36)(28,40)(29,33)(31,37)(39,42)(45,48)(46,47)Polycyclic presentation:
2^9|b^2=i,d^2=fgi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6640331);