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 | #3013 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29)(20,26)(21,28)(22,27)(23,25)(24,31)(30,43)(32,42)(33,41)(34,39)(35,40)(36,37)(38,45)(44,47)(46,48);(17,19,23,32)(18,24,25,37)(20,30,34,46)(21,31,22,36)(26,38,39,47)(27,29,28,42)(33,44,35,45)(40,48,41,43);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(18,25)(24,37)(26,39)(27,28)(29,42)(38,47)(40,41)(43,48);(3,9)(6,12)(10,15)(13,16)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,38)(25,39)(27,41)(28,40)(29,43)(31,45)(32,46)(36,44)(37,47)(42,48);(17,21,23,22)(18,27,25,28)(19,31,32,36)(20,33,34,35)(24,29,37,42)(26,40,39,41)(30,44,46,45)(38,48,47,43)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,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=di,e^d=eh110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252210);