Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,30)(18,22)(19,23)(24,32)(25,33)(26,34)(27,35)(28,37)(29,38)(31,36)(39,42)(40,43)(41,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,27)(14,20,23,35)(16,24,29,41)(17,26,18,25)(21,32,38,44)(22,33,30,34)(28,39,31,40)(36,43,37,42);(2,4)(13,16,19,29)(14,21,23,38)(15,24,27,41)(17,31,18,28)(20,32,35,44)(22,37,30,36)(25,39,26,40)(33,42,34,43);(2,4)(5,7,8,11)(6,10,9,12)(15,25)(17,18)(20,33)(22,30)(24,39)(26,27)(28,31)(32,42)(34,35)(36,37)(40,41)(43,44);(6,9)(10,12)(13,17,19,18)(14,22,23,30)(15,26,27,25)(16,28,29,31)(20,33,35,34)(21,36,38,37)(24,39,41,40)(32,43,44,42)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423421);