Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2)(3,4)(6,9)(10,12)(17,29)(22,31)(25,36)(27,40)(33,41)(34,39)(38,43)(42,44);(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(21,26)(22,31)(24,35)(25,36)(34,39)(38,43);(1,2,3,4)(13,14,18,19)(15,20,23,30)(16,26,28,21)(17,22,29,31)(24,37,35,32)(25,33,36,41)(27,39,40,34)(38,44,43,42);(1,2)(3,4)(13,15)(14,20)(16,24)(17,25)(18,23)(19,30)(21,32)(22,33)(26,37)(27,38)(28,35)(29,36)(31,41)(34,42)(39,44)(40,43);(13,16,18,28)(14,21,19,26)(15,24,23,35)(17,27,29,40)(20,32,30,37)(22,34,31,39)(25,38,36,43)(33,42,41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,25)(16,27)(18,29)(19,31)(20,33)(21,34)(23,36)(24,38)(26,39)(28,40)(30,41)(32,42)(35,43)(37,44)Polycyclic presentation:
2^9|c^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581367);