Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | |
2036 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)(18,24,34,39)(28,40,42,36)(31,43,41,37)(33,44,35,38);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,21,25,26)(16,17,27,20)(18,31,34,41)(22,32,30,23)(24,37,39,43)(28,35,42,33)(36,38,40,44);(1,2,3,4)(6,9)(10,12)(13,16,20,15,19,27,17,29)(14,22,23,21,25,30,32,26)(18,31,35,28,34,41,33,42)(24,37,38,36,39,43,44,40);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,32)(15,27,29,16)(18,33,34,35)(21,22,26,30)(24,38,39,44)(28,41,42,31)(36,37,40,43);(13,18,20,35,19,34,17,33)(14,24,23,38,25,39,32,44)(15,28,27,41,29,42,16,31)(21,36,30,43,26,40,22,37)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=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724259);