Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,21)(17,30)(18,23)(19,26)(20,28)(22,32)(25,40)(27,38)(29,36)(31,35)(33,44)(34,42)(37,43)(39,41);(1,2)(3,4)(13,15,18,27)(14,19,23,35)(16,25,28,41)(17,31,32,26)(20,33,21,34)(22,38,30,24)(29,44,43,42)(36,40,37,39);(2,4)(6,9)(10,12)(13,16,18,28)(14,20,23,21)(15,25,27,41)(17,29,32,43)(19,33,35,34)(22,36,30,37)(24,39,38,40)(26,42,31,44);(2,4)(13,16)(14,21)(15,25)(17,29)(18,28)(19,34)(20,23)(22,37)(24,40)(26,42)(27,41)(30,36)(31,44)(32,43)(33,35)(38,39);(5,7,8,11)(6,10,9,12)(16,28)(20,21)(25,41)(29,43)(33,34)(36,37)(39,40)(42,44)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361020);