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 |
| prime radical | C2×#3277 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2096 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)(5,6,9,11)(7,12,8,10)(13,14,19,25)(15,26,29,21)(16,22,27,36)(17,31,20,23)(18,24,34,41)(28,42,43,37)(30,38,33,40)(32,44,35,39);(5,6)(7,10)(8,12)(9,11)(13,15,19,29)(14,21,25,26)(16,17,27,20)(18,30,34,33)(22,23,36,31)(24,40,41,38)(28,35,43,32)(37,39,42,44);(1,2,3,4)(13,16,20,15,19,27,17,29)(14,22,23,26,25,36,31,21)(18,33,35,43,34,30,32,28)(24,40,39,37,41,38,44,42);(5,7,9,8)(6,10,11,12)(13,17,19,20)(14,23,25,31)(15,27,29,16)(18,32,34,35)(21,36,26,22)(24,39,41,44)(28,33,43,30)(37,40,42,38);(5,8,9,7)(6,10,11,12)(13,18,20,35,19,34,17,32)(14,24,23,39,25,41,31,44)(15,28,27,33,29,43,16,30)(21,37,22,38,26,42,36,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=f,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724030);