Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #42880 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 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)(8,12)(9,11)(13,14,19,23)(15,20,28,37)(16,21,32,40)(17,22,33,18)(24,35,41,43)(25,27,36,26)(29,38,42,44)(30,31,39,34);(5,7,10,6)(8,12,9,11)(13,15,19,28)(14,20,23,37)(16,29,32,42)(17,25,33,36)(18,26,22,27)(21,38,40,44)(24,34,41,31)(30,43,39,35);(2,4)(13,16,19,32)(14,21,23,40)(15,24,28,41)(17,30,33,39)(18,34,22,31)(20,35,37,43)(25,38,36,44)(26,29,27,42);(5,8,10,9)(6,11,7,12)(13,17)(14,22)(15,25)(16,30)(18,23)(19,33)(20,27)(21,31)(24,38)(26,37)(28,36)(29,43)(32,39)(34,40)(35,42)(41,44);(5,9,10,8)(6,11,7,12)(15,26)(17,33)(18,22)(20,25)(24,29)(27,28)(30,39)(31,34)(35,38)(36,37)(41,42)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584068);