Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,9,19,32,17,33,18,31)(4,20,24,10,22,35,23,36)(5,25,27,11,26,37,28,38)(15,39,42,29,43,48,41,56)(16,44,45,30,47,54,46,57)(21,50,51,58,52,34,53,49)(40,60,62,64,61,55,63,59);(1,3,6,19,8,17,7,18)(2,9,14,31,12,33,13,32)(4,15,24,41,22,43,23,42)(5,16,27,46,26,47,28,45)(10,29,20,48,36,56,35,39)(11,30,25,54,38,57,37,44)(21,40,53,62,52,61,51,63)(34,55,49,64,50,60,58,59);(1,4)(2,10)(3,15)(5,21)(6,24)(7,23)(8,22)(9,29)(11,34)(12,36)(13,35)(14,20)(16,40)(17,43)(18,42)(19,41)(25,49)(26,52)(27,53)(28,51)(30,55)(31,48)(32,39)(33,56)(37,58)(38,50)(44,59)(45,63)(46,62)(47,61)(54,64)(57,60);(1,5)(2,11)(3,16)(4,21)(6,27)(7,28)(8,26)(9,30)(10,34)(12,38)(13,37)(14,25)(15,40)(17,47)(18,45)(19,46)(20,49)(22,52)(23,51)(24,53)(29,55)(31,54)(32,44)(33,57)(35,58)(36,50)(39,59)(41,62)(42,63)(43,61)(48,64)(56,60)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72341351, #72341353, #72341363, … |
| maximal normal subgroup of | #72341351, #72341353, #72341363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299911);