Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,22)(16,21)(17,31)(18,34)(19,24)(20,27)(23,33)(25,39)(26,40)(28,36)(29,35)(30,43)(32,44)(37,41)(38,42);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,26,33,17)(14,20,23,22,24,31,34,40)(16,25,30,29,28,41,42,32)(21,35,38,39,36,44,43,37);(1,2,4,3)(6,9)(10,11)(13,16)(14,21)(15,25)(17,32)(18,30)(19,28)(20,35)(22,39)(23,38)(24,36)(26,41)(27,29)(31,44)(33,42)(34,43)(37,40);(1,3,4,2)(5,7,8,12)(6,10,9,11)(15,26)(16,28)(17,27)(20,31)(21,36)(22,40)(30,42)(38,43);(13,17,33,26,19,27,18,15)(14,22,34,20,24,40,23,31)(16,29,42,25,28,32,30,41)(21,37,43,44,36,39,38,35)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504197);