Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1941 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)(8,9)(11,12)(13,14,18,25,19,24,20,26)(15,27,30,39,31,40,32,21)(16,33,17,23,34,22,35,42)(28,41,44,37,36,38,29,43);(1,2)(3,4)(13,15,19,31)(14,21,24,39)(16,29,34,44)(17,36,35,28)(18,32,20,30)(22,41,33,38)(23,37,42,43)(25,40,26,27);(2,4)(6,9)(10,12)(13,16,20,35,19,34,18,17)(14,22,26,23,24,33,25,42)(15,28,32,29,31,36,30,44)(21,37,40,41,39,43,27,38);(2,4)(13,17,18,34,19,35,20,16)(14,23,25,22,24,42,26,33)(15,29,30,28,31,44,32,36)(21,38,27,43,39,41,40,37);(5,7,8,11)(6,10,9,12)(14,24)(21,39)(22,33)(23,42)(25,26)(27,40)(37,43)(38,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363688);