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 | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2182 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)(8,10)(9,12)(13,14,19,24)(15,20,27,34)(16,21,29,36)(17,22,18,23)(25,32,41,44)(26,33,30,37)(28,35,31,38)(39,42,40,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(16,28)(17,18)(21,35)(22,23)(25,39)(26,30)(29,31)(32,42)(33,37)(36,38)(40,41)(43,44);(2,4)(13,15,19,27)(14,20,24,34)(16,25,29,41)(17,30,18,26)(21,32,36,44)(22,37,23,33)(28,39,31,40)(35,42,38,43);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,25)(17,28)(18,31)(19,29)(20,32)(22,35)(23,38)(24,36)(26,40)(27,41)(30,39)(33,43)(34,44)(37,42);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,22,24,23)(15,26,27,30)(16,28,29,31)(20,33,34,37)(21,35,36,38)(25,40,41,39)(32,43,44,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=fi,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, 7429648);