Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,39)(27,34)(28,36)(29,37)(30,38)(32,35)(33,41)(40,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,23,35)(16,25,29,42)(17,26,18,32)(21,33,37,43)(22,34,31,24)(28,40,30,41)(36,39,38,44);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,28)(19,29)(20,33)(22,38)(23,37)(24,39)(26,41)(27,42)(31,36)(32,40)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,37,38)(25,40,42,41)(33,39,43,44);(7,11)(10,12)(14,23)(20,35)(21,37)(22,31)(24,34)(33,43)(36,38)(39,44)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440649);