Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(5,6)(7,18)(8,23)(9,15)(10,16)(11,31)(12,17)(13,21)(14,27)(19,49)(20,48)(22,37)(24,35)(25,55)(26,42)(28,60)(29,44)(30,46)(32,38)(33,41)(34,45)(36,53)(39,51)(40,58)(43,54)(47,52)(50,66)(56,61)(57,64)(59,63)(62,67)(65,68);(2,3)(5,7)(6,13)(8,24)(9,28)(10,21)(11,22)(12,32)(14,39)(15,43)(16,18)(17,37)(19,27)(20,30)(23,47)(25,50)(26,52)(29,54)(31,38)(33,51)(34,53)(35,42)(36,46)(40,61)(41,49)(44,60)(45,48)(55,62)(56,59)(57,67)(58,68)(63,65)(64,66);(1,2,4,3)(5,8,11,27,10,26,12,33)(6,14,17,23,16,41,31,42)(7,19,22,52,21,51,32,24)(9,25,30,58,29,57,34,59)(13,35,37,39,18,47,38,49)(15,40,45,55,44,63,46,64)(20,50,54,68,53,67,28,56)(36,61,43,62,48,65,60,66);(1,3,4,2)(5,9,10,29)(6,15,16,44)(7,20,21,53)(8,25,26,57)(11,30,12,34)(13,36,18,48)(14,40,41,63)(17,45,31,46)(19,50,51,67)(22,54,32,28)(23,55,42,64)(24,56,52,68)(27,58,33,59)(35,61,47,65)(37,43,38,60)(39,62,49,66);(7,21)(13,18)(19,51)(20,53)(22,32)(24,52)(28,54)(35,47)(36,48)(37,38)(39,49)(43,60)(50,67)(56,68)(61,65)(62,66)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351497);