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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,23)(18,21)(19,34)(20,25)(24,28)(27,41)(29,37)(30,35)(31,40)(32,36)(33,39)(38,43)(42,44);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,18,20,28,17,32)(14,21,23,26,25,36,34,24)(16,27,29,30,31,43,33,42)(22,35,39,41,40,44,37,38);(6,10)(7,11)(13,16)(14,22)(15,27)(17,33)(18,30)(19,29)(20,31)(21,35)(23,39)(24,38)(25,40)(26,41)(28,43)(32,42)(34,37)(36,44);(1,2,3,4)(6,10)(9,12)(13,17,20,19)(14,23,25,34)(15,18,28,32)(16,29,31,33)(21,24,36,26)(22,37,40,39)(27,42,43,30)(35,41,44,38);(13,18,19,28,20,32,17,15)(14,24,23,21,25,26,34,36)(16,30,29,43,31,42,33,27)(22,38,39,35,40,41,37,44)Polycyclic presentation:
2^9|b^2=f,d^2=h,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518969);