Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,28,36)(16,29,32,37)(17,21,18,23)(22,31,38,30)(25,39,42,43)(26,33,27,35)(34,41,44,40);(6,9)(10,12)(13,15,19,28)(14,20,24,36)(16,25,32,42)(17,26,18,27)(21,33,23,35)(22,34,38,44)(29,39,37,43)(30,40,31,41);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(17,18)(20,33)(22,37)(23,24)(26,27)(29,38)(30,31)(34,43)(35,36)(39,44)(40,41);(13,16)(14,22)(15,25)(17,30)(18,31)(19,32)(20,34)(21,37)(23,29)(24,38)(26,40)(27,41)(28,42)(33,43)(35,39)(36,44);(13,17,19,18)(14,23,24,21)(15,26,28,27)(16,30,32,31)(20,35,36,33)(22,29,38,37)(25,40,42,41)(34,39,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,h^a=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536446);