Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,21)(17,22)(18,33)(19,24)(23,26)(25,34)(27,35)(28,36)(29,37)(30,38)(31,42)(32,39)(40,43)(41,44);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,28)(17,18)(19,27)(21,36)(22,33)(23,37)(24,35)(25,30)(26,29)(31,32)(34,38)(39,42)(40,41)(43,44);(1,2,3,4)(13,16,29,41,19,32,17,30)(14,21,37,44,24,39,22,38)(15,25,18,31,27,40,26,28)(20,34,33,42,35,43,23,36);(6,10)(9,11)(13,17,19,29)(14,22,24,37)(15,18,27,26)(16,30,32,41)(20,33,35,23)(21,38,39,44)(25,31,40,28)(34,42,43,36);(7,12)(9,11)(13,18,29,27,19,26,17,15)(14,23,37,20,24,33,22,35)(16,31,41,40,32,28,30,25)(21,36,44,34,39,42,38,43)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721673);