Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61447 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9,6,3,7,11)(4,8,13,15,12,10,14,16)(17,18)(19,29)(20,26)(21,28)(22,27)(23,25)(24,31)(30,43)(32,42)(33,39)(34,41)(35,40)(36,37)(38,46)(44,48)(45,47);(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(18,25)(20,33)(24,37)(27,28)(29,42)(30,44)(34,35)(45,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20,23,33)(18,26,25,39)(19,30,32,44)(21,35,22,34)(24,38,37,47)(27,41,28,40)(29,43,42,48)(31,46,36,45);(4,12)(8,10)(13,14)(15,16)(17,21,23,22)(18,27,25,28)(19,31,32,36)(20,34,33,35)(24,29,37,42)(26,40,39,41)(30,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=eh,f^a=fh,f^b=fh,d^c=dh,e^d=ehi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253252);