Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,19,25)(15,26,29,21)(16,30,27,22)(17,32,20,23)(18,24,35,41)(28,42,43,37)(31,40,34,38)(33,44,36,39);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,21,25,26)(16,17,27,20)(18,31,35,34)(22,32,30,23)(24,38,41,40)(28,36,43,33)(37,39,42,44);(1,2,3,4)(6,9)(10,12)(13,16,20,15,19,27,17,29)(14,22,23,21,25,30,32,26)(18,34,36,43,35,31,33,28)(24,40,39,42,41,38,44,37);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,32)(15,27,29,16)(18,33,35,36)(21,22,26,30)(24,39,41,44)(28,34,43,31)(37,38,42,40);(13,18,20,36,19,35,17,33)(14,24,23,39,25,41,32,44)(15,28,27,34,29,43,16,31)(21,37,30,40,26,42,22,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724301);