Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,28)(24,36)(25,30)(26,39)(27,31)(29,37)(32,46)(33,42)(34,43)(35,49)(38,44)(40,45)(41,48)(47,52)(50,51);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23)(22,28)(24,32)(25,34)(26,35)(27,33)(29,41)(30,43)(31,42)(36,46)(37,48)(38,47)(39,49)(40,50)(44,52)(45,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,29)(23,32)(25,37)(26,40)(27,38)(28,41)(30,44)(31,45)(33,47)(34,48)(35,50)(36,39)(42,51)(43,52)(46,49);(6,12)(10,13)(11,20)(18,19)(23,33)(28,42)(32,47)(34,35)(41,51)(43,49)(46,52)(48,50);(21,25,27,26)(22,30,31,39)(23,34,33,35)(24,37,38,40)(28,43,42,49)(29,44,45,36)(32,48,47,50)(41,52,51,46)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281562);