Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| 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 | #3449 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,14,19,30,18,39,17,9)(4,10,21,33,22,34,27,32)(5,11,24,35,25,36,23,50)(15,40,42,28,43,37,47,53)(16,38,46,54,45,59,44,29)(20,31,48,56,49,57,51,55)(41,60,61,52,62,58,63,64);(1,3)(2,9,13,30)(4,15)(5,16)(6,17)(7,19)(8,18)(10,28,34,53)(11,29,36,54)(12,14,26,39)(20,41)(21,42)(22,43)(23,44)(24,46)(25,45)(27,47)(31,52,57,64)(32,40,33,37)(35,38,50,59)(48,61)(49,62)(51,63)(55,60,56,58);(1,4)(2,10)(3,15)(5,20)(6,27)(7,21)(8,22)(9,28)(11,31)(12,33)(13,34)(14,37)(16,41)(17,47)(18,43)(19,42)(23,51)(24,48)(25,49)(26,32)(29,52)(30,53)(35,56)(36,57)(38,58)(39,40)(44,63)(45,62)(46,61)(50,55)(54,64)(59,60);(1,5,7,24,8,25,6,23)(2,11,12,35,13,36,26,50)(3,16,19,46,18,45,17,44)(4,20,21,48,22,49,27,51)(9,29,14,38,30,54,39,59)(10,31,33,56,34,57,32,55)(15,41,42,61,43,62,47,63)(28,52,37,58,53,64,40,60)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,f^b=fi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72282115, #72285525, #72285688, … |
| maximal normal subgroup of | #72282115, #72285525, #72285688, … |
| maximal quotient of | C2×#354310, #72343056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261506);