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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,47)(32,41)(33,42)(34,40)(35,45)(36,43)(37,46)(38,44)(39,48)(49,52)(50,51);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,25,33)(22,27,28,42)(24,35,36,50)(26,31,38,34)(29,45,43,51)(30,47,44,40)(32,39,49,37)(41,48,52,46);(1,2,3,4)(22,28)(23,31)(26,38)(27,40)(29,43)(32,35)(33,34)(37,39)(41,51)(42,47)(45,52)(49,50);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,36)(29,43)(32,49)(35,50)(37,39)(41,52)(45,51)(46,48);(7,15)(10,13)(14,20)(18,19)(21,24,25,36)(22,29,28,43)(23,32,33,49)(26,39,38,37)(27,41,42,52)(30,48,44,46)(31,35,34,50)(40,51,47,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=f,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, 6591381);