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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1712 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)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,24)(23,31,35,28)(25,36,38,29)(27,30,40,37)(32,34,44,41)(33,42,48,45)(39,50,51,43)(46,47,52,49);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,26,35)(22,28,24,31)(25,33,38,48)(27,41,40,34)(29,42,36,45)(30,44,37,32)(39,47,51,49)(43,52,50,46);(2,4)(21,24,26,22)(23,32,35,44)(25,36,38,29)(27,30,40,37)(28,41,31,34)(33,46,48,52)(39,43,51,50)(42,47,45,49);(6,11)(10,13)(12,19)(18,20)(21,25)(22,29)(23,33)(24,36)(26,38)(27,39)(28,42)(30,43)(31,45)(32,46)(34,49)(35,48)(37,50)(40,51)(41,47)(44,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,34)(25,38)(27,40)(28,32)(29,36)(30,37)(31,44)(33,47)(35,41)(42,52)(45,46)(48,49)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=fi,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, 6580821);