Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358021 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3499 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18,23,25)(19,28,32,42)(20,26,33,39)(21,27,22,36)(24,30,37,31)(29,43,44,48)(34,40,35,41)(38,46,47,45);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,21,30,23,32,22,31)(18,24,36,42,25,37,27,28)(20,29,35,46,33,44,34,45)(26,38,40,48,39,47,41,43);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(18,25)(20,33)(24,37)(27,36)(28,42)(29,44)(34,35)(45,46);(3,10)(6,13)(9,15)(12,16)(17,20)(18,26)(19,29)(21,35)(22,34)(23,33)(24,38)(25,39)(27,41)(28,43)(30,46)(31,45)(32,44)(36,40)(37,47)(42,48);(2,7)(6,13)(8,14)(12,16)(17,21,23,22)(18,27,25,36)(19,30,32,31)(20,34,33,35)(24,28,37,42)(26,40,39,41)(29,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=dhi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265236);