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 | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2618 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)(7,19)(8,25)(9,15)(10,16)(11,17)(12,18)(13,23)(14,28)(20,49)(21,48)(22,38)(24,37)(26,35)(27,55)(29,41)(30,60)(31,44)(32,45)(33,42)(34,46)(36,53)(39,51)(40,57)(43,54)(47,52)(50,66)(56,61)(58,63)(59,64)(62,67)(65,68);(2,3)(5,7)(6,13)(8,26)(9,30)(10,24)(11,23)(12,22)(14,39)(15,43)(16,38)(17,19)(18,37)(20,28)(21,31)(25,47)(27,50)(29,52)(32,54)(33,51)(34,53)(35,41)(36,44)(40,61)(42,49)(45,60)(46,48)(55,62)(56,59)(57,68)(58,67)(63,66)(64,65);(1,2,4,3)(5,8,10,28,11,29,12,33)(6,14,18,25,17,42,16,41)(7,20,24,52,23,51,22,26)(9,27,31,57,32,58,34,59)(13,35,37,39,19,47,38,49)(15,40,46,55,45,64,44,63)(21,50,54,68,53,67,30,56)(36,61,43,62,48,65,60,66);(1,3,4,2)(5,9,11,32)(6,15,17,45)(7,21,23,53)(8,27,29,58)(10,31,12,34)(13,36,19,48)(14,40,42,64)(16,44,18,46)(20,50,51,67)(22,30,24,54)(25,55,41,63)(26,56,52,68)(28,57,33,59)(35,61,47,65)(37,43,38,60)(39,62,49,66);(5,10,11,12)(6,16,17,18)(7,22,23,24)(8,28,29,33)(9,31,32,34)(13,37,19,38)(14,41,42,25)(15,44,45,46)(20,26,51,52)(21,30,53,54)(27,57,58,59)(35,39,47,49)(36,43,48,60)(40,63,64,55)(50,56,67,68)(61,62,65,66)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=f,e^2=i,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, 6351500);