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 | #36078 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,25)(15,27,31,21)(16,32,29,44)(17,35,28,41)(18,26,20,24)(22,36,40,34)(23,33,42,37)(30,39,38,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,33)(17,36)(18,30)(19,31)(20,38)(22,35)(23,32)(24,39)(25,27)(26,43)(28,34)(29,37)(40,41)(42,44);(13,16)(14,22)(15,28)(17,31)(18,34)(19,29)(20,36)(21,23)(24,32)(25,40)(26,44)(27,42)(30,37)(33,38)(35,43)(39,41);(2,4)(13,17,19,28)(14,23,25,42)(15,29,31,16)(18,37,20,33)(21,22,27,40)(24,35,26,41)(30,34,38,36)(32,39,44,43);(6,10)(9,11)(13,18,19,20)(14,24,25,26)(15,30,31,38)(16,34,29,36)(17,33,28,37)(21,39,27,43)(22,32,40,44)(23,41,42,35)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421875);