Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,21)(16,22)(17,23)(18,33)(19,24)(20,25)(26,35)(27,36)(28,38)(29,39)(30,40)(31,42)(32,41)(34,37)(43,44);(1,2,3,4)(6,9)(10,11)(13,15)(14,21)(16,30)(17,26)(18,28)(19,34)(20,29)(22,40)(23,35)(24,37)(25,39)(27,32)(31,43)(33,38)(36,41)(42,44);(5,7,8,12)(6,10,9,11)(13,16,19,17,20,32,18,31)(14,22,24,23,25,41,33,42)(15,26,28,30,29,43,34,27)(21,35,38,40,39,44,37,36);(13,17,19,32,20,31,18,16)(14,23,24,41,25,42,33,22)(15,27,28,26,29,30,34,43)(21,36,38,35,39,40,37,44);(13,18,20,19)(14,24,25,33)(15,28,29,34)(16,31,32,17)(21,37,39,38)(22,23,41,42)(26,30,43,27)(35,36,44,40)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6374007);