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×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,3,4)(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);(2,4)(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);(5,8,12,33,11,29,10,28)(6,14,16,41,17,42,18,25)(7,20,22,26,23,51,24,52)(9,27,34,59,32,58,31,57)(13,35,38,49,19,47,37,39)(15,40,44,63,45,64,46,55)(21,50,30,56,53,67,54,68)(36,61,60,66,48,65,43,62);(5,9)(6,15)(7,21)(8,27)(10,31)(11,32)(12,34)(13,36)(14,40)(16,44)(17,45)(18,46)(19,48)(20,50)(22,30)(23,53)(24,54)(25,55)(26,56)(28,57)(29,58)(33,59)(35,61)(37,43)(38,60)(39,62)(41,63)(42,64)(47,65)(49,66)(51,67)(52,68);(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=gi,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, 6350152);