Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×#3179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,23,20,24,18,16)(15,25,28,21,29,30,27,37)(17,22,35,40,33,39,34,31)(26,41,32,36,42,38,43,44);(2,4)(13,15,20,29)(14,21,24,37)(16,30,23,25)(17,32,33,43)(18,28,19,27)(22,38,39,41)(26,34,42,35)(31,44,40,36);(6,9)(10,12)(13,16,18,24,20,23,19,14)(15,21,27,25,29,37,28,30)(17,31,34,39,33,40,35,22)(26,36,43,41,42,44,32,38);(13,17,18,34,20,33,19,35)(14,22,16,31,24,39,23,40)(15,26,27,43,29,42,28,32)(21,36,25,41,37,44,30,38);(5,7,8,11)(6,10,9,12)(17,33)(22,39)(26,42)(31,40)(32,43)(34,35)(36,44)(38,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374835);