Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,27,40)(16,21,30,37)(17,23,18,22)(20,33,35,28)(26,41,29,43)(31,39,32,38)(34,44,36,42);(7,8)(10,11)(13,15)(14,20)(16,29)(17,28)(18,33)(19,27)(21,36)(22,25)(23,40)(24,35)(26,30)(31,44)(32,42)(34,37)(38,43)(39,41);(2,4)(5,7,9,8)(6,10,12,11)(13,16,19,30)(14,21,24,37)(15,26,27,29)(17,31,18,32)(20,34,35,36)(22,38,23,39)(25,41,40,43)(28,42,33,44);(5,8,9,7)(6,11,12,10)(15,27)(16,30)(20,35)(21,37)(25,40)(28,33)(31,32)(38,39);(6,12)(10,11)(13,17,19,18)(14,22,24,23)(15,28,27,33)(16,31,30,32)(20,25,35,40)(21,38,37,39)(26,42,29,44)(34,41,36,43)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394644);