Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2094 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)(7,11)(8,10)(9,12)(13,14)(15,26)(16,32)(17,25)(18,37)(19,24)(20,30)(21,35)(22,36)(23,33)(27,39)(28,40)(29,41)(31,44)(34,42)(38,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,33)(17,28)(18,35)(19,30)(21,23)(22,40)(24,26)(25,44)(27,34)(29,38)(31,36)(32,37)(39,41)(42,43);(2,4)(7,12)(9,11)(13,16,17,34,19,35,36,29)(14,21,25,41,24,32,22,42)(15,27,31,33,30,38,28,18)(20,39,40,23,26,43,44,37);(6,10)(9,11)(13,17,19,36)(14,22,24,25)(15,28,30,31)(16,34,35,29)(18,38,33,27)(20,40,26,44)(21,42,32,41)(23,43,37,39);(13,18,36,38,19,33,17,27)(14,23,22,39,24,37,25,43)(15,29,28,35,30,34,31,16)(20,41,44,21,26,42,40,32)Polycyclic presentation:
2^9|c^2=fhi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717329);