Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22030 |
| prime radical | #3025 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,15,34,48,9,16,35,49,6,13,32,46)(3,10,19,31,18,37,53,5,12,27,43,26,29,44,61,28)(4,20,25,55,24,41,50,47,23,45,33,62,30,11,38,42)(17,39,51,22,52,36,63,60,40,64,54,58,21,56,57,59);(1,3,7,19,8,18,34,53,9,12,35,43,6,29,32,61)(2,10,14,31,15,37,48,5,16,27,49,26,13,44,46,28)(4,21,25,57,24,17,50,51,23,52,33,63,30,40,38,54)(11,39,42,22,20,36,55,60,41,64,47,58,45,56,62,59);(1,4)(2,11)(3,17)(5,22)(6,30)(7,33)(8,24)(9,23)(10,36)(12,40)(13,45)(14,47)(15,20)(16,41)(18,52)(19,54)(21,29)(25,35)(26,60)(27,56)(28,58)(31,59)(32,50)(34,38)(37,64)(39,44)(42,49)(43,51)(46,55)(48,62)(53,57)(61,63);(1,5)(2,12)(3,16)(4,22)(6,31)(7,27)(8,26)(9,28)(10,35)(11,40)(13,18)(14,43)(15,29)(17,41)(19,49)(20,21)(23,58)(24,60)(25,36)(30,59)(32,37)(33,56)(34,44)(38,39)(42,54)(45,52)(46,53)(47,51)(48,61)(50,64)(55,57)(62,63)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ceh,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224961, #72320755, C2×#496805, … |
| maximal normal subgroup of | #72224961, #72320755, C2×#496805, … |
| maximal quotient of | C2×#496805 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404001);