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 | #45101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1832 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)(7,11)(8,12)(9,10)(13,14,18,24)(15,21,27,20)(16,22,30,40)(17,31,19,23)(25,37,43,34)(26,35,32,36)(28,44,42,38)(29,41,33,39);(1,2)(3,4)(6,10)(11,12)(13,15,18,27)(14,20,24,21)(16,28,30,42)(17,26,19,32)(22,38,40,44)(23,35,31,36)(25,33,43,29)(34,39,37,41);(2,4)(5,7,9,8)(6,11,10,12)(13,15)(14,21)(16,25)(17,32)(18,27)(19,26)(20,24)(22,37)(23,35)(28,29)(30,43)(31,36)(33,42)(34,40)(38,39)(41,44);(2,4)(13,16,18,30)(14,22,24,40)(15,25,27,43)(17,29,19,33)(20,34,21,37)(23,39,31,41)(26,42,32,28)(35,38,36,44);(5,8,9,7)(6,11,10,12)(13,17,18,19)(14,23,24,31)(15,26,27,32)(16,29,30,33)(20,35,21,36)(22,39,40,41)(25,42,43,28)(34,38,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,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, 7427145);