Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(24,36)(28,44)(34,38)(37,50)(39,45)(43,46)(48,51)(49,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,22,26,30)(23,31,33,47)(24,37,36,50)(25,40,32,29)(27,41,42,35)(28,45,44,39)(34,46,38,43)(48,52,51,49);(2,4)(21,23,32,41,26,33,25,35)(22,27,40,47,30,42,29,31)(24,38,45,49,36,34,39,52)(28,46,50,48,44,43,37,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(27,42)(31,47)(34,38)(35,41)(43,46)(48,51)(49,52);(7,15)(10,13)(14,20)(18,19)(21,24,32,45,26,36,25,39)(22,28,40,50,30,44,29,37)(23,34,41,52,33,38,35,49)(27,43,47,51,42,46,31,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=hi,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,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715848);