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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14)(15,26)(16,30)(17,35)(18,23)(19,25)(20,24)(21,28)(22,33)(27,42)(29,37)(31,41)(32,40)(34,39)(36,44)(38,43);(1,2)(3,4)(6,10)(11,12)(13,15,18,28)(14,21,23,26)(16,31,32,44)(17,19,29,20)(22,38,39,42)(24,37,25,35)(27,34,43,33)(30,36,40,41);(5,7,9,8)(6,11,10,12)(13,16,19,33,18,32,20,34)(14,22,24,30,23,39,25,40)(15,27,17,31,28,43,29,44)(21,36,35,38,26,41,37,42);(2,4)(13,17,20,15,18,29,19,28)(14,21,25,37,23,26,24,35)(16,31,34,27,32,44,33,43)(22,36,40,42,39,41,30,38);(5,8,9,7)(6,11,10,12)(14,23)(16,32)(21,26)(24,25)(27,43)(31,44)(33,34)(35,37)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=gi,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472053);