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 |
| prime radical | #60990 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,43)(30,39)(32,44)(33,47)(34,41)(35,42)(36,48)(38,46)(40,45);(2,6)(7,9)(8,13)(14,15)(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);(2,7)(5,12)(6,9)(8,14)(11,16)(13,15)(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)(2,8)(4,10)(5,11)(6,13)(7,14)(9,15)(12,16)(20,34)(25,41)(29,33)(35,36)(38,40)(42,48)(43,47)(45,46);(3,10)(8,13)(11,16)(14,15)(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|b^2=fhi,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, 6263465);