Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,28)(19,30,34,24)(20,26,36,44)(21,29,23,27)(25,38,43,35)(31,40,32,46)(33,41,37,45)(39,48,42,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,34)(18,24,28,30)(20,32,36,31)(21,37,23,33)(25,42,43,39)(26,40,44,46)(27,45,29,41)(35,47,38,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,43)(19,31,34,32)(21,35,23,38)(24,39,30,42)(26,29,44,27)(33,47,37,48)(40,45,46,41);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,32)(21,38)(22,36)(23,35)(24,40)(25,27)(28,44)(29,43)(30,46)(31,34)(33,47)(37,48)(39,41)(42,45);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,33,34,37)(20,35,36,38)(24,41,30,45)(25,26,43,44)(31,47,32,48)(39,40,42,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7208829);