Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496254 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21750 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #3500 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3500 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14,18,42)(4,19,24,54)(5,12,27,45)(6,11)(7,13)(8,17,34,28)(9,23)(10,26,41,61)(15,48)(16,47)(20,55)(21,38)(22,39,29,36)(25,32)(30,40)(31,56,62,57)(33,60)(35,50,46,49)(37,59)(43,58)(44,52)(51,63)(53,64);(1,3)(2,8)(4,20)(5,25)(6,28)(7,18)(9,35)(10,32)(11,42)(12,44)(13,34)(14,40)(15,22)(16,27)(17,30)(19,48)(21,57)(23,56)(24,53)(26,52)(29,51)(31,37)(33,45)(36,55)(38,46)(39,64)(41,47)(43,62)(49,59)(50,58)(54,63)(60,61);(1,4,7,24)(2,9,13,38)(3,15,18,51)(5,23,27,21)(6,29,30,22)(8,31,34,62)(10,39,41,36)(11,43,40,37)(12,19,45,54)(14,46,42,35)(16,50,25,49)(17,53,28,20)(26,58,61,59)(32,63,47,48)(33,55,44,64)(52,56,60,57);(1,5,7,27)(2,10)(3,16,18,25)(4,21,24,23)(6,26,30,61)(8,32)(9,36)(11,12)(13,41)(14,33)(15,49,51,50)(17,52,28,60)(19,37)(20,56,53,57)(22,58,29,59)(31,48)(34,47)(35,55)(38,39)(40,45)(42,44)(43,54)(46,64)(62,63)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=ei,g^c=gh,f^d=fi,g^d=gi,g^f=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403451);