Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,35)(24,32)(25,30)(26,34)(27,40)(28,31)(29,36)(33,37)(38,44)(39,42)(41,43);(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(16,27)(18,24)(20,33)(21,36)(25,37)(28,29)(30,35)(31,40)(34,42)(38,41);(1,2,3,4)(6,9)(10,12)(13,15,24,37,17,25,18,23)(14,20,32,33,19,30,22,35)(16,26,27,41,28,39,29,38)(21,34,40,43,31,42,36,44);(5,7,8,11)(6,10,9,12)(14,19)(15,25)(21,31)(22,32)(23,37)(26,39)(36,40)(38,41);(13,16,18,29,17,28,24,27)(14,21,22,36,19,31,32,40)(15,26,23,38,25,39,37,41)(20,34,35,44,30,42,33,43)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724260);