Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,28)(25,38)(26,34)(27,39)(29,35)(30,37)(31,36)(33,40)(41,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,26,19,27,18,28)(14,20,32,39,23,34,22,24)(16,25,31,42,30,41,29,40)(21,33,35,43,37,44,36,38);(1,2,3,4)(6,9)(10,12)(13,16,19,30)(14,21,23,37)(15,25,27,41)(17,31,18,29)(20,33,34,44)(22,36,32,35)(24,38,39,43)(26,42,28,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,28)(16,29,30,31)(20,24,34,39)(21,35,37,36)(25,40,41,42)(33,43,44,38);(7,11)(10,12)(15,27)(20,34)(24,39)(25,41)(26,28)(33,44)(38,43)(40,42)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465451);