Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | #60453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(5,6,12,18)(7,19,24,37)(8,25,28,43)(9,29,32,15)(10,17,11,16)(13,22,38,23)(14,33,40,27)(20,35,49,61)(21,41,51,58)(26,53,57,64)(30,36,50,62)(31,42,34,44)(39,56,59,55)(45,48,52,47)(46,65,67,60)(54,63,66,68);(1,2)(3,4)(5,7)(6,13)(8,20)(9,30)(10,22)(11,23)(12,24)(14,35)(15,41)(16,37)(17,19)(18,38)(21,31)(25,52)(26,54)(27,47)(28,49)(29,58)(32,50)(33,48)(34,51)(36,42)(39,63)(40,61)(43,45)(44,62)(46,55)(53,60)(56,67)(57,66)(59,68)(64,65);(2,4)(5,8,12,28)(6,14,18,40)(7,20,24,49)(9,26,32,57)(10,27,11,33)(13,35,38,61)(15,39,29,59)(16,25,17,43)(19,45,37,52)(21,46,51,67)(22,47,23,48)(30,54,50,66)(31,55,34,56)(36,60,62,65)(41,63,58,68)(42,53,44,64);(5,9,10,31,12,32,11,34)(6,15,16,42,18,29,17,44)(7,21,23,30,24,51,22,50)(8,26,27,55,28,57,33,56)(13,36,19,41,38,62,37,58)(14,39,25,53,40,59,43,64)(20,46,48,54,49,67,47,66)(35,60,45,63,61,65,52,68);(5,10,12,11)(6,16,18,17)(7,22,24,23)(8,27,28,33)(9,31,32,34)(13,37,38,19)(14,25,40,43)(15,42,29,44)(20,47,49,48)(21,50,51,30)(26,55,57,56)(35,52,61,45)(36,58,62,41)(39,53,59,64)(46,66,67,54)(60,68,65,63)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409351);