Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2685 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,21)(17,29)(18,20)(19,25)(22,30)(24,36)(26,35)(27,33)(28,41)(31,39)(32,38)(34,40)(37,43)(42,44);(5,7)(6,9)(8,11)(10,12)(13,15,17,25,18,26,30,39)(14,19,22,23,20,31,29,35)(16,24,28,38,27,37,40,44)(21,32,34,36,33,42,41,43);(1,2,3,4)(7,11)(9,12)(14,20)(19,31)(21,33)(22,29)(23,35)(32,42)(34,41)(36,43);(6,10)(9,12)(13,16)(14,21)(15,24)(17,28)(18,27)(19,32)(20,33)(22,34)(23,36)(25,38)(26,37)(29,41)(30,40)(31,42)(35,43)(39,44);(16,27)(21,33)(24,37)(28,40)(32,42)(34,41)(36,43)(38,44)Polycyclic presentation:
2^9|b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340998);