Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,21,18,22,31,37)(15,23,25,33,26,36,30,19)(16,27,29,43,28,20,34,35)(24,39,41,32,40,38,44,42);(1,2)(3,4)(13,15,18,26)(14,19,22,33)(16,24,28,40)(17,30,31,25)(20,32,27,42)(21,36,37,23)(29,44,34,41)(35,39,43,38);(2,4)(6,9)(10,12)(13,16,18,28)(14,20,22,27)(15,24,26,40)(17,29,31,34)(19,32,33,42)(21,35,37,43)(23,38,36,39)(25,41,30,44);(2,4)(16,28)(20,27)(24,40)(29,34)(32,42)(35,43)(38,39)(41,44);(5,7,8,11)(6,10,9,12)(16,28)(20,27)(24,40)(29,34)(32,42)(35,43)(38,39)(41,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361549);