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 |
| prime radical | #45196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,21)(16,22)(17,29)(18,20)(19,23)(24,34)(25,36)(26,32)(27,38)(28,33)(30,35)(31,37)(39,44)(40,43)(41,42);(1,2)(3,4)(6,10)(11,12)(14,20)(16,27)(17,19)(21,32)(22,37)(24,39)(25,30)(28,31)(33,38)(34,43)(40,41)(42,44);(2,4)(5,7,9,8)(6,11,10,12)(13,15)(14,21)(16,24)(17,30)(18,26)(19,25)(20,32)(22,34)(23,36)(27,39)(28,41)(29,35)(31,40)(33,42)(37,43)(38,44);(2,4)(13,16,18,28)(14,22,20,33)(15,24,26,41)(17,27,19,31)(21,34,32,42)(23,37,29,38)(25,40,30,39)(35,44,36,43);(5,8,9,7)(6,11,10,12)(13,17,18,19)(14,23,20,29)(15,25,26,30)(16,27,28,31)(21,35,32,36)(22,37,33,38)(24,40,41,39)(34,44,42,43)Polycyclic presentation:
2^9|c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7427153);