Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#3301 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 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)(7,10)(8,9)(11,12)(14,19)(16,27)(20,33)(22,31)(23,38)(24,28)(29,41)(40,43);(6,9)(7,11)(13,14)(15,23)(16,21)(17,19)(18,31)(20,26)(22,30)(24,39)(25,38)(27,34)(28,42)(29,44)(32,33)(35,40)(36,43)(37,41);(2,4)(13,15,17,25)(14,20,19,33)(16,28,27,24)(18,32,30,26)(21,36,34,35)(22,38,31,23)(29,40,41,43)(37,39,44,42);(6,9)(10,12)(13,16,17,27)(14,21,19,34)(15,24,25,28)(18,29,30,41)(20,35,33,36)(22,37,31,44)(23,39,38,42)(26,40,32,43);(5,7,8,11)(6,10,9,12)(14,22)(15,25)(18,30)(19,31)(20,23)(21,37)(24,28)(29,41)(33,38)(34,44)(35,39)(36,42)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579965);