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 | #32885 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,26)(17,22)(18,31)(19,23)(21,24)(25,34)(27,40)(28,33)(29,37)(30,38)(32,36)(35,39)(41,44)(42,43);(6,9)(10,12)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,29,30,25)(21,33,36,26)(22,37,38,34)(27,41,42,39)(35,40,44,43);(13,16,19,28)(14,21,23,36)(15,24,18,32)(17,27,30,42)(20,33,31,26)(22,35,38,44)(25,39,29,41)(34,43,37,40);(2,4)(13,17)(14,22)(15,25)(16,27)(18,29)(19,30)(20,34)(21,35)(23,38)(24,39)(26,40)(28,42)(31,37)(32,41)(33,43)(36,44);(5,7,8,11)(6,10,9,12)(17,30)(22,38)(25,29)(27,42)(34,37)(35,44)(39,41)(40,43)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536539);