Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1968 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,29)(17,23)(18,24)(19,25)(21,28)(22,32)(26,37)(27,36)(30,44)(31,41)(33,38)(34,40)(35,43)(39,42);(1,2)(3,4)(6,9)(10,11)(13,15,18,28)(14,20,24,21)(16,30,32,42)(17,27,19,33)(22,39,29,44)(23,36,25,38)(26,34,43,31)(35,41,37,40);(2,4)(13,15)(14,21)(16,26)(17,33)(18,28)(19,27)(20,24)(22,37)(23,36)(25,38)(29,35)(30,31)(32,43)(34,42)(39,41)(40,44);(2,4)(13,16,18,32)(14,22,24,29)(15,26,28,43)(17,31,19,34)(20,35,21,37)(23,40,25,41)(27,42,33,30)(36,44,38,39);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,23,24,25)(15,27,28,33)(16,31,32,34)(20,36,21,38)(22,40,29,41)(26,42,43,30)(35,44,37,39)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425651);