Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,20)(16,21)(17,23)(18,24)(19,25)(22,32)(26,34)(27,39)(28,36)(29,37)(30,38)(31,40)(33,41)(35,43)(42,44);(5,7)(6,10)(8,11)(9,12)(13,15,19,30)(14,20,25,38)(16,27,32,43)(17,29,18,28)(21,39,22,35)(23,37,24,36)(26,33,42,31)(34,41,44,40);(2,4)(5,8,9,10)(6,7,11,12)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,25)(23,40)(24,41)(27,28)(29,43)(30,42)(35,37)(36,39)(38,44);(2,4)(6,11)(7,12)(13,16)(14,22)(15,27)(17,33)(18,31)(19,32)(20,35)(21,25)(23,40)(24,41)(26,28)(29,42)(30,43)(34,37)(36,44)(38,39);(13,17,19,18)(14,23,25,24)(15,28,30,29)(16,31,32,33)(20,36,38,37)(21,40,22,41)(26,27,42,43)(34,39,44,35)Polycyclic presentation:
2^9|b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408417);