Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356241 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18,23,27)(19,28,32,44)(20,25,34,41)(21,26,22,37)(24,30,39,31)(29,43,33,47)(35,42,36,48)(38,45,40,46);(17,19,21,30,23,32,22,31)(18,24,37,44,27,39,26,28)(20,33,35,46,34,29,36,45)(25,40,48,43,41,38,42,47);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,20)(18,25)(19,29)(21,35)(22,36)(23,34)(24,38)(26,42)(27,41)(28,43)(30,45)(31,46)(32,33)(37,48)(39,40)(44,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(20,34)(25,41)(29,33)(35,36)(38,40)(42,48)(43,47)(45,46);(3,9)(7,14)(10,15)(13,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,35,34,36)(24,28,39,44)(25,42,41,48)(29,45,33,46)(38,43,40,47)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263475);