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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14,17,22)(15,24,27,19)(16,29,31,40)(18,21,33,23)(20,34,38,32)(25,42,43,37)(26,39,28,36)(30,41,44,35);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,27)(14,19,22,24)(16,30,31,44)(18,26,33,28)(20,37,38,42)(21,36,23,39)(25,34,43,32)(29,35,40,41);(5,7,9,8)(6,11,12,10)(13,16)(14,20)(15,25)(17,31)(18,34)(19,35)(21,29)(22,38)(23,40)(24,41)(26,30)(27,43)(28,44)(32,33)(36,42)(37,39);(2,4)(14,21)(15,26)(18,33)(20,29)(22,23)(25,30)(27,28)(32,34)(36,39)(37,42)(38,40)(43,44);(5,8,9,7)(6,11,12,10)(14,22)(16,31)(19,24)(21,23)(25,43)(30,44)(32,34)(36,39)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7427928);