Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470368 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,26,30)(23,31,33,45)(24,28,35,42)(25,37,38,29)(27,39,41,34)(32,44,46,52)(36,49,48,43)(40,50,51,47);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,25,34,26,33,38,39)(22,27,29,31,30,41,37,45)(24,32,36,47,35,46,48,50)(28,40,43,44,42,51,49,52);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24)(22,28)(23,32)(25,36)(26,35)(27,40)(29,43)(30,42)(31,44)(33,46)(34,47)(37,49)(38,48)(39,50)(41,51)(45,52);(7,13)(10,17)(14,19)(18,20)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(44,52)(47,50);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(23,33)(27,41)(31,45)(32,46)(34,39)(40,51)(44,52)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405933);