Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,30,23)(20,24,32,40)(22,26,34,42)(28,45,46,37)(29,43,35,38)(31,48,47,39)(33,41,36,44);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,25,27)(20,31,32,47)(22,35,34,29)(24,39,40,48)(26,43,42,38)(28,36,46,33)(37,44,45,41);(10,14)(13,15)(17,20)(18,24)(19,28)(21,32)(22,36)(23,37)(25,40)(26,44)(27,45)(29,47)(30,46)(31,35)(33,34)(38,48)(39,43)(41,42);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,25)(20,32)(23,27)(26,42)(28,46)(31,47)(33,36)(38,43);(2,5)(6,8)(11,16)(13,15)(19,29)(22,34)(23,38)(26,42)(27,43)(28,47)(30,35)(31,46)(33,36)(37,48)(39,45)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606330);