Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14807 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1267 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)(3,9)(4,19,21,36)(5,24,25,38)(6,12)(7,14)(8,13)(10,22,35,23)(11,27,37,26)(15,40,42,49)(16,44,45,55)(17,31)(18,32)(20,51)(28,39)(29,43,33,47)(30,46,58,48)(34,54)(41,57)(50,52)(53,56)(59,62)(60,63)(61,64);(1,3,6,17,8,18,7,28)(2,9,12,31,13,32,14,39)(4,15,23,47,21,42,22,43)(5,16,27,48,25,45,26,46)(10,33,36,40,35,29,19,49)(11,30,24,55,37,58,38,44)(20,41,52,61,54,62,53,63)(34,59,56,60,51,57,50,64);(1,4)(2,10)(3,15)(5,20)(6,23)(7,22)(8,21)(9,29)(11,34)(12,36)(13,35)(14,19)(16,41)(17,47)(18,42)(24,56)(25,54)(26,53)(27,52)(28,43)(30,57)(31,49)(32,33)(37,51)(38,50)(39,40)(44,60)(45,62)(46,63)(48,61)(55,64)(58,59);(1,5,6,27,8,25,7,26)(2,11,12,24,13,37,14,38)(3,16,17,48,18,45,28,46)(4,20,23,52,21,54,22,53)(9,30,31,55,32,58,39,44)(10,34,36,56,35,51,19,50)(15,41,47,61,42,62,43,63)(29,57,49,64,33,59,40,60)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285493, C2×#400817, #72285784, … |
| maximal normal subgroup of | #72285493, C2×#400817, #72285784, … |
| maximal quotient of | C2×#400817 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308013);