Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,26)(23,28,34,33)(24,29,37,41)(25,30,40,39)(31,42,47,35)(32,43,50,49)(36,44,38,45)(46,52,48,51);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,28)(24,35)(25,32)(26,33)(27,34)(29,31)(30,43)(36,51)(37,42)(38,52)(39,49)(40,50)(41,47)(44,46)(45,48);(2,4)(21,24,27,37)(22,29,26,41)(23,31,34,47)(25,38,40,36)(28,42,33,35)(30,45,39,44)(32,48,50,46)(43,51,49,52);(6,11)(10,13)(12,19)(18,20)(21,25,27,40)(22,30,26,39)(23,32,34,50)(24,36,37,38)(28,43,33,49)(29,44,41,45)(31,46,47,48)(35,51,42,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,26)(23,33)(28,34)(29,41)(30,39)(31,35)(32,49)(42,47)(43,50)(44,45)(46,51)(48,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580648);