Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,18)(17,28)(19,22)(20,27)(21,30)(24,33)(25,34)(26,35)(29,32)(31,42)(36,43)(37,41)(38,40)(39,44);(6,9)(7,11)(13,15)(14,20)(16,23)(17,25)(18,27)(19,33)(21,34)(22,35)(24,29)(26,32)(28,37)(30,41)(31,44)(36,39)(38,42)(40,43);(2,4)(13,16,18,14)(15,24,27,35)(17,21,30,28)(19,22,32,29)(20,33,23,26)(25,39,41,38)(31,42,43,36)(34,40,37,44);(6,9)(10,12)(13,17,18,30)(14,21,16,28)(15,25,27,41)(19,31,32,43)(20,34,23,37)(22,36,29,42)(24,38,35,39)(26,40,33,44);(5,7,8,11)(6,10,9,12)(14,16)(15,26)(19,32)(20,35)(21,28)(23,24)(25,40)(27,33)(31,43)(34,39)(37,38)(41,44)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580013);