Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,12)(9,11)(13,14)(15,23)(16,27)(17,21)(18,32)(19,25)(20,30)(22,31)(24,40)(26,35)(28,44)(29,38)(33,37)(34,43)(36,41)(39,42);(2,3)(5,7,10,6)(8,12,9,11)(13,15)(14,19)(16,28)(17,25)(18,26)(20,36)(21,23)(22,35)(24,29)(27,40)(30,43)(31,42)(32,39)(33,41)(34,37)(38,44);(1,2,4,3)(13,16,17,29)(14,20,21,37)(15,24,25,28)(18,33,31,30)(19,34,23,36)(22,38,32,27)(26,43,42,41)(35,40,39,44);(5,8,10,9)(6,11,7,12)(14,21)(15,25)(20,37)(22,32)(24,28)(26,42)(27,38)(41,43);(1,3,4,2)(5,9,10,8)(6,11,7,12)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718776);