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 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2037 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)(13,14,19,24)(15,25,17,31)(16,21,29,36)(18,34,26,23)(20,33,22,28)(27,39,32,43)(30,42,40,37)(35,44,38,41);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,26,33,19,17,18,28)(14,20,34,31,24,22,23,25)(16,27,40,44,29,32,30,41)(21,35,42,43,36,38,37,39);(1,2,4,3)(6,9)(10,11)(13,16)(14,21)(15,27)(17,32)(18,30)(19,29)(20,35)(22,38)(23,37)(24,36)(25,39)(26,40)(28,41)(31,43)(33,44)(34,42);(1,3,4,2)(5,7,8,12)(6,10,9,11)(15,17)(16,29)(20,22)(21,36)(25,31)(28,33)(30,40)(37,42);(13,17,18,33,19,15,26,28)(14,22,23,31,24,20,34,25)(16,27,30,41,29,32,40,44)(21,35,37,39,36,38,42,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,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, 7504218);