Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 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)(5,6,8,9)(7,11,12,10)(13,14,19,24)(15,20,27,35)(16,21,30,38)(17,22,18,23)(25,33,42,44)(26,34,31,39)(28,36,41,43)(29,37,32,40);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,24,35)(16,28,30,41)(17,26,18,31)(21,36,38,43)(22,34,23,39)(25,32,42,29)(33,40,44,37);(2,4)(6,9)(10,11)(13,16,19,30)(14,21,24,38)(15,25,27,42)(17,29,18,32)(20,33,35,44)(22,37,23,40)(26,41,31,28)(34,43,39,36);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,39)(21,37,38,40)(25,41,42,28)(33,43,44,36);(15,26)(17,18)(20,34)(22,23)(25,41)(27,31)(28,42)(29,32)(33,43)(35,39)(36,44)(37,40)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578270);