Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2045 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)(13,14)(15,24)(16,21)(17,31)(18,23)(19,22)(20,28)(25,39)(26,35)(27,34)(29,38)(30,37)(32,36)(33,41)(40,44)(42,43);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,34)(23,24)(29,40)(30,41)(31,35)(32,42)(36,44)(37,39)(38,43);(2,4)(13,16,18,30)(14,21,23,37)(15,25,28,41)(17,32,19,29)(20,33,24,39)(22,38,31,36)(26,40,27,42)(34,43,35,44);(2,4)(5,7,8,11)(6,10,9,12)(15,26)(17,19)(20,34)(22,31)(24,35)(25,40)(27,28)(29,32)(33,43)(36,38)(39,44)(41,42);(6,9)(10,12)(13,17,18,19)(14,22,23,31)(15,27,28,26)(16,29,30,32)(20,34,24,35)(21,36,37,38)(25,40,41,42)(33,44,39,43)Polycyclic presentation:
2^9|c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423287);