Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 | #37445 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2083 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)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,35)(16,30,33,37)(17,22,18,23)(20,28,36,27)(21,31,38,32)(26,34,42,44)(39,40,43,41);(5,7)(6,9)(8,11)(10,12)(13,15,18,28,19,29,17,27)(14,20,22,35,24,36,23,25)(16,26,32,41,33,42,31,40)(21,34,30,43,38,44,37,39);(2,4)(7,11)(9,12)(13,16,17,31,19,33,18,32)(14,21,23,37,24,38,22,30)(15,26,27,40,29,42,28,41)(20,34,25,39,36,44,35,43);(7,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,40,42,41)(34,43,44,39);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,33,31)(20,35,36,25)(21,37,38,30)(26,41,42,40)(34,39,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378133);