Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,29,36)(16,30,32,21)(17,24,19,22)(20,27,37,28)(26,41,43,38)(31,44,42,35)(33,39,34,40);(1,2,3,4)(13,15,19,27,18,29,17,28)(14,20,24,36,23,37,22,25)(16,31,34,43,32,42,33,26)(21,38,40,44,30,41,39,35);(6,9)(10,12)(13,16,18,32)(14,21,23,30)(15,26,29,43)(17,34,19,33)(20,35,37,44)(22,40,24,39)(25,38,36,41)(27,42,28,31);(15,27)(17,19)(20,36)(22,24)(25,37)(26,42)(28,29)(31,43)(33,34)(35,41)(38,44)(39,40);(5,7,8,11)(6,10,9,12)(16,32)(21,30)(26,43)(31,42)(33,34)(35,44)(38,41)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6362146);