Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | |
2024 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)(15,23)(16,27)(17,19)(18,22)(20,25)(21,30)(24,34)(26,35)(28,37)(29,38)(31,41)(32,33)(36,39)(40,44)(42,43);(5,6)(7,10)(8,9)(11,12)(14,19)(15,24)(16,28)(18,32)(20,34)(21,37)(23,36)(25,39)(27,41)(30,31)(35,38)(40,42);(1,2,3,4)(6,9)(10,12)(13,15,18,24,17,25,32,39)(14,20,22,36,19,23,33,34)(16,29,31,42,30,26,28,40)(21,38,37,43,27,35,41,44);(5,7,8,11)(6,10,9,12)(14,19)(15,25)(21,27)(22,33)(24,39)(26,29)(37,41)(40,42);(13,16,18,31,17,30,32,28)(14,21,22,37,19,27,33,41)(15,26,24,40,25,29,39,42)(20,35,36,44,23,38,34,43)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,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724321);