Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,34)(24,37)(25,40)(26,42)(27,33)(28,31)(29,36)(30,39)(32,44)(35,46)(38,47)(41,48)(43,49)(45,52)(50,51);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,29)(24,25)(26,43)(27,36)(28,39)(30,40)(31,37)(32,49)(33,34)(35,41)(38,44)(42,47)(45,51)(46,52)(48,50);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,25,23,27,39,28,36)(22,30,40,29,33,37,31,34)(26,38,41,35,44,43,45,51)(32,47,52,46,42,49,48,50);(6,12)(10,13)(11,18)(17,19)(21,25,27,28)(22,31,33,40)(23,24,36,39)(26,41,44,45)(29,37,34,30)(32,48,42,52)(35,38,51,43)(46,49,50,47);(7,15)(10,13)(14,20)(17,19)(21,26,25,41,27,44,28,45)(22,32,40,52,33,42,31,48)(23,35,39,43,36,51,24,38)(29,46,37,49,34,50,30,47)Polycyclic presentation:
2^9|c^2=fh,d^2=i,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=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714518);