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×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,25,28,34)(16,22,33,37)(17,23,19,24)(20,29,36,26)(27,38,42,43)(30,35,41,44)(31,39,32,40);(1,2,3,4)(13,15,19,29,18,28,17,26)(14,20,23,34,21,36,24,25)(16,30,31,42,33,41,32,27)(22,38,40,44,37,43,39,35);(6,9)(10,12)(14,21)(15,26)(17,19)(20,34)(22,37)(25,36)(27,30)(28,29)(31,32)(35,43)(38,44)(41,42);(13,16,18,33)(14,22,21,37)(15,27,28,42)(17,31,19,32)(20,35,36,44)(23,39,24,40)(25,38,34,43)(26,30,29,41);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,21,24)(15,26,28,29)(16,31,33,32)(20,34,36,25)(22,39,37,40)(27,30,42,41)(35,43,44,38)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,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, 6362111);