Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,28,20)(16,30,34,21)(17,32,36,22)(19,24,37,33)(26,39,43,35)(27,40,44,31)(29,42,38,41);(5,7,9,8)(6,10,12,11)(13,15,18,28)(14,20,23,25)(16,31,34,40)(17,35,36,39)(19,38,37,29)(21,27,30,44)(22,26,32,43)(24,42,33,41);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,26)(17,24)(18,34)(19,32)(20,39)(22,37)(23,30)(25,35)(27,42)(28,43)(29,40)(31,38)(33,36)(41,44);(5,8,9,7)(6,10,12,11)(16,32)(17,21)(19,37)(22,34)(24,33)(26,40)(27,35)(29,38)(30,36)(31,43)(39,44)(41,42);(13,17,18,36)(14,22,23,32)(15,27,28,44)(16,33,34,24)(19,30,37,21)(20,40,25,31)(26,41,43,42)(29,39,38,35)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484375);