Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | |
3316 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)(17,23)(18,31)(19,22)(20,26)(25,38)(27,35)(28,37)(29,36)(30,41)(32,34)(33,40)(39,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,22,34)(16,25,28,39)(17,26,18,32)(21,33,37,44)(23,35,31,24)(29,40,30,42)(36,43,41,38);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,28)(20,33)(22,37)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(34,44)(35,43);(5,7)(6,10)(8,11)(9,12)(14,22)(16,28)(20,34)(23,31)(24,35)(25,39)(29,30)(40,42);(7,11)(10,12)(13,17,19,18)(14,23,22,31)(15,26,27,32)(16,29,28,30)(20,35,34,24)(21,36,37,41)(25,40,39,42)(33,43,44,38)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440616);