Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1603 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)(17,18)(19,24)(20,26)(21,28)(22,27)(23,29)(25,34)(30,46)(31,39)(32,45)(33,41)(35,44)(36,40)(37,43)(38,48)(42,47);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19,23,33)(18,24,29,41)(20,31,35,30)(21,36,22,32)(25,42,43,38)(26,39,44,46)(27,45,28,40)(34,47,37,48);(9,10)(11,14)(12,13)(15,16)(17,20,23,35)(18,25,29,43)(19,30,33,31)(21,34,22,37)(24,38,41,42)(26,28,44,27)(32,47,36,48)(39,45,46,40);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20)(18,26)(19,31)(21,37)(22,34)(23,35)(24,39)(25,27)(28,43)(29,44)(30,33)(32,47)(36,48)(38,40)(41,46)(42,45);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,27,29,28)(19,32,33,36)(20,34,35,37)(24,40,41,45)(25,26,43,44)(30,47,31,48)(38,39,42,46)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7388683);