Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14,19,18)(15,20,25,32)(16,21,27,28)(17,22,31,33)(23,34,39,43)(24,35,41,29)(26,36,30,37)(38,44,40,42);(2,3)(6,10)(11,12)(13,15,19,25)(14,20,18,32)(16,23,27,39)(17,29,31,35)(21,34,28,43)(22,24,33,41)(26,42,30,44)(36,38,37,40);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,23)(17,30)(18,28)(19,27)(20,34)(22,37)(24,40)(25,39)(26,31)(29,44)(32,43)(33,36)(35,42)(38,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,24)(16,26)(18,33)(19,31)(20,35)(21,36)(23,38)(25,41)(27,30)(28,37)(29,32)(34,44)(39,40)(42,43);(6,10)(11,12)(16,27)(17,31)(21,28)(22,33)(23,39)(24,41)(29,35)(34,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468827);