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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,22)(16,21)(17,30)(18,34)(19,25)(20,24)(23,33)(26,36)(27,37)(28,39)(29,38)(31,40)(32,41)(35,42)(43,44);(2,3)(6,10)(11,12)(13,15)(14,21)(16,19)(17,31)(18,27)(20,29)(22,25)(23,40)(24,37)(26,32)(28,35)(30,44)(33,43)(34,38)(36,42)(39,41);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,19,15)(14,22,25,21)(17,26,33,28)(18,27,20,29)(23,36,30,39)(24,37,34,38)(31,35,43,32)(40,41,44,42);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,19,33)(14,23,25,30)(15,26,16,28)(18,32,20,35)(21,36,22,39)(24,41,34,42)(27,43,29,31)(37,40,38,44);(6,10)(11,12)(13,18,19,20)(14,24,25,34)(15,27,16,29)(17,32,33,35)(21,37,22,38)(23,41,30,42)(26,43,28,31)(36,40,39,44)Polycyclic presentation:
2^9|c^2=fgi,d^2=fgi,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,d^c=di,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468778);