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×#3275 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(17,29)(22,38)(26,41)(28,31)(30,37)(35,44)(39,43)(40,42);(6,9)(7,11)(13,14,19,23)(15,24,27,34)(16,21,18,32)(17,30,29,37)(20,33,36,25)(22,31,38,28)(26,35,41,44)(39,40,43,42);(2,4)(5,7,8,11)(6,10,9,12)(13,15,19,27)(14,20,23,36)(16,25,18,33)(17,26,29,41)(21,34,32,24)(22,35,38,44)(28,40,31,42)(30,39,37,43);(6,9)(10,12)(13,16,19,18)(14,21,23,32)(15,25,27,33)(17,31,29,28)(20,34,36,24)(22,30,38,37)(26,42,41,40)(35,39,44,43);(13,17,19,29)(14,22,23,38)(15,26,27,41)(16,28,18,31)(20,35,36,44)(21,37,32,30)(24,39,34,43)(25,40,33,42)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,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, 6576311);