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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,40)(32,39)(33,41)(34,38)(35,42)(36,44)(37,43)(45,51)(46,49)(47,50)(48,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,25,33)(22,27,28,41)(24,32,35,47)(26,31,37,34)(29,39,42,50)(30,40,43,38)(36,45,48,46)(44,51,52,49);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(22,28)(23,31)(26,37)(27,38)(29,42)(32,45)(33,34)(36,48)(39,49)(40,41)(46,47)(50,51);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24)(22,29)(23,32)(25,35)(26,36)(27,39)(28,42)(30,44)(31,45)(33,47)(34,46)(37,48)(38,49)(40,51)(41,50)(43,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,31)(24,35)(26,37)(27,40)(29,42)(30,43)(32,46)(33,34)(38,41)(39,49)(45,47)(50,51)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=f,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, 6595020);