Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | |
1891 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)(13,14,18,23,19,24,33,35)(15,25,17,22,27,40,32,20)(16,28,31,43,30,21,36,39)(26,41,29,34,42,37,44,38);(1,2)(3,4)(13,15,19,27)(14,20,24,22)(16,29,30,44)(17,18,32,33)(21,37,28,41)(23,40,35,25)(26,36,42,31)(34,43,38,39);(2,4)(6,9)(10,12)(13,16,18,31,19,30,33,36)(14,21,23,39,24,28,35,43)(15,26,17,29,27,42,32,44)(20,34,25,37,22,38,40,41);(2,4)(13,17,18,27,19,32,33,15)(14,22,23,40,24,20,35,25)(16,29,31,42,30,44,36,26)(21,38,39,41,28,34,43,37);(5,7,8,11)(6,10,9,12)(16,30)(21,28)(26,42)(29,44)(31,36)(34,38)(37,41)(39,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363822);