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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1999 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)(15,25)(16,21)(17,22)(18,23)(19,24)(20,28)(26,39)(27,34)(29,37)(30,35)(31,36)(32,38)(33,41)(40,44)(42,43);(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,29)(17,26)(18,28)(19,27)(21,34)(22,33)(23,25)(24,37)(30,42)(31,41)(32,40)(35,44)(36,39)(38,43);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(17,31)(18,19)(22,36)(23,24)(26,40)(27,29)(33,43)(34,37)(39,44)(41,42);(5,8,9,7)(6,10,12,11)(13,17,18,31)(14,22,23,36)(15,26,28,41)(16,30,19,32)(20,33,25,39)(21,35,24,38)(27,40,29,42)(34,44,37,43);(14,23)(15,27)(16,19)(17,32)(20,34)(22,35)(25,37)(28,29)(30,31)(33,39)(36,38)(40,42)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=f,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, 6484366);