Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35634 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,27)(16,22)(17,23)(18,24)(19,25)(20,26)(21,30)(28,41)(29,44)(31,43)(32,40)(33,42)(34,37)(35,38)(36,39);(1,2,3,4)(13,15,19,30)(14,21,25,27)(16,31,28,36)(17,32,34,33)(18,29,20,35)(22,39,41,43)(23,42,37,40)(24,38,26,44);(6,9)(10,12)(13,16,19,28)(14,22,25,41)(15,17,30,34)(18,33,20,32)(21,37,27,23)(24,42,26,40)(29,36,35,31)(38,43,44,39);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,28)(16,30)(18,31)(19,34)(20,36)(21,22)(24,43)(25,37)(26,39)(27,41)(29,33)(32,35)(38,40)(42,44);(13,18,19,20)(14,24,25,26)(15,29,30,35)(16,32,28,33)(17,31,34,36)(21,38,27,44)(22,40,41,42)(23,43,37,39)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356274);