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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,25)(15,27,30,21)(16,32,28,44)(17,36,38,42)(18,26,20,24)(22,35,43,34)(23,37,39,33)(29,40,31,41);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,33)(17,34)(18,29)(19,30)(20,31)(22,42)(23,44)(24,40)(25,27)(26,41)(28,37)(32,39)(35,38)(36,43);(13,16,19,28)(14,22,25,43)(15,17,30,38)(18,35,20,34)(21,39,27,23)(24,44,26,32)(29,33,31,37)(36,40,42,41);(2,4)(5,7,9,8)(6,11,12,10)(13,17)(14,23)(15,28)(16,30)(18,37)(19,38)(20,33)(21,22)(24,42)(25,39)(26,36)(27,43)(29,35)(31,34)(32,40)(41,44);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,24,25,26)(15,29,30,31)(16,34,28,35)(17,37,38,33)(21,40,27,41)(22,32,43,44)(23,42,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429801);