Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2282 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)(7,10,11,12)(13,14,18,25,20,26,19,24)(15,27,29,21,30,37,17,23)(16,22,33,41,35,42,34,40)(28,43,31,36,44,38,32,39);(6,9)(10,12)(13,15)(14,21)(16,31)(17,18)(19,29)(20,30)(22,38)(23,26)(24,37)(25,27)(28,33)(32,35)(34,44)(36,41)(39,40)(42,43);(1,2,3,4)(13,16,19,34,20,35,18,33)(14,22,24,40,26,42,25,41)(15,28,17,32,30,44,29,31)(21,36,27,43,23,39,37,38);(5,7,8,11)(6,10,9,12)(13,17,19,30,20,29,18,15)(14,23,24,37,26,21,25,27)(16,32,34,44,35,31,33,28)(22,39,40,38,42,36,41,43);(13,18,20,19)(14,24,26,25)(15,17,30,29)(16,33,35,34)(21,37,23,27)(22,40,42,41)(28,32,44,31)(36,38,39,43)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367226);