Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #25390 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1609 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,31,44)(20,32,35,47)(21,36,29,25)(23,37,39,28)(24,40,42,33)(30,41,34,38)(43,48,46,45);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(20,33)(21,29)(23,27)(24,32)(25,36)(30,34)(31,37)(35,40)(38,41)(39,44)(42,47);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,29,37,22,31,21,28)(18,23,36,44,26,39,25,27)(20,34,40,45,35,30,33,48)(24,41,47,43,42,38,32,46);(9,11,12,16)(10,14,13,15)(17,20,21,33,22,35,29,40)(18,24,25,32,26,42,36,47)(19,30,28,45,31,34,37,48)(23,38,27,43,39,41,44,46);(1,4,5,3)(2,7,6,8)(20,35)(24,42)(30,34)(32,47)(33,40)(38,41)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=i,c^2=fhi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504614);