Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #31001 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1547 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,27)(20,31)(21,35)(22,25)(23,29)(24,33)(26,36)(28,43)(30,42)(32,41)(34,40)(37,45)(38,47)(39,46)(44,48);(2,5)(3,7)(10,13)(11,15)(17,19,21,29,22,30,36,47)(18,23,26,27,25,38,35,42)(20,32,34,45,33,28,44,46)(24,39,40,43,31,37,48,41);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,31)(26,40)(27,41)(29,46)(30,32)(35,48)(36,44)(38,39)(42,43)(45,47);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,33)(21,36)(23,27)(24,31)(26,35)(28,45)(30,47)(32,46)(37,43)(38,42)(39,41);(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,33)(26,35)(29,47)(34,44)(37,39)(41,43)Polycyclic presentation:
2^9|b^2=fgi,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7069294);