Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 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)(13,14,17,24,18,23,19,25)(15,26,28,37,29,38,30,20)(16,22,32,41,33,21,34,40)(27,39,43,35,44,42,31,36);(1,2)(3,4)(13,15,18,29)(14,20,23,37)(16,31,33,43)(17,30,19,28)(21,39,22,42)(24,38,25,26)(27,32,44,34)(35,40,36,41);(2,4)(6,9)(10,12)(13,16,19,34,18,33,17,32)(14,21,25,41,23,22,24,40)(15,27,30,31,29,44,28,43)(20,35,38,39,37,36,26,42);(2,4)(13,16,17,32,18,33,19,34)(14,22,24,41,23,21,25,40)(15,27,28,43,29,44,30,31)(20,36,26,39,37,35,38,42);(5,7,8,11)(6,10,9,12)(14,23)(20,37)(21,22)(24,25)(26,38)(35,36)(39,42)(40,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363699);