Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(21,22)(23,31)(24,29)(25,28)(26,30)(27,34)(32,43)(33,49)(35,41)(36,52)(37,44)(38,47)(39,45)(40,48)(42,51)(46,50);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23,25,34)(22,27,28,31)(24,36,37,50)(26,32,39,35)(29,46,44,52)(30,41,45,43)(33,38,51,40)(42,47,49,48);(2,4)(23,32)(26,39)(27,41)(30,45)(31,43)(33,50)(34,35)(36,51)(38,40)(42,52)(46,49)(47,48);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,28)(24,37)(27,31)(30,45)(33,51)(36,50)(38,40)(41,43);(7,15)(10,13)(14,20)(18,19)(21,24,25,37)(22,29,28,44)(23,33,34,51)(26,40,39,38)(27,42,31,49)(30,48,45,47)(32,50,35,36)(41,52,43,46)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579523);