Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #386 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | C4×#386 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,18,32,19,31,17,9)(4,20,22,10,23,34,28,35)(5,11,25,36,26,37,24,52)(15,40,42,48,44,29,43,38)(16,39,46,56,47,55,45,30)(21,49,50,33,51,57,53,58)(41,60,61,64,63,54,62,59);(1,3)(2,9,13,32)(4,15)(5,16)(6,17)(7,18)(8,19)(10,29,35,40)(11,30,37,56)(12,14,27,31)(20,48,34,38)(21,41)(22,42)(23,44)(24,45)(25,46)(26,47)(28,43)(33,54,58,60)(36,39,52,55)(49,64,57,59)(50,61)(51,63)(53,62);(1,4)(2,10)(3,15)(5,21)(6,28)(7,22)(8,23)(9,29)(11,33)(12,34)(13,35)(14,38)(16,41)(17,43)(18,42)(19,44)(20,27)(24,53)(25,50)(26,51)(30,54)(31,48)(32,40)(36,57)(37,58)(39,59)(45,62)(46,61)(47,63)(49,52)(55,64)(56,60);(1,5,7,25,8,26,6,24)(2,11,12,36,13,37,27,52)(3,16,18,46,19,47,17,45)(4,21,22,50,23,51,28,53)(9,30,14,39,32,56,31,55)(10,33,34,57,35,58,20,49)(15,41,42,61,44,63,43,62)(29,54,38,59,40,60,48,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,f^b=fi,e^c=ei,f^c=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72281077, #72285525, #72285688, … |
| maximal normal subgroup of | #72281077, #72285525, #72285688, … |
| maximal quotient of | C2×#354322, #72342990 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261518);