Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1260 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)(3,16)(4,22)(5,26)(6,15)(7,13)(8,14)(9,18)(10,24)(11,23)(12,30)(17,34)(19,43)(20,49)(21,50)(25,45)(27,35)(28,46)(29,39)(31,44)(32,41)(33,54)(36,56)(37,55)(38,63)(40,58)(42,59)(47,52)(48,51)(53,61)(57,60)(62,64);(1,3,8,21)(2,9,14,36)(4,17,25,53)(5,27,31,55)(6,20,7,19)(10,38,41,33)(11,34,40,61)(12,43,39,49)(13,37,15,35)(16,44,50,26)(18,29,56,30)(22,48,45,62)(23,54,58,63)(24,57,32,52)(28,51,59,64)(42,47,46,60);(1,4)(2,10)(3,17)(5,23)(6,32)(7,24)(8,25)(9,33)(11,39)(12,40)(13,45)(14,41)(15,22)(16,47)(18,51)(19,57)(20,52)(21,53)(26,46)(27,54)(28,30)(29,59)(31,58)(34,43)(35,62)(36,38)(37,48)(42,44)(49,61)(50,60)(55,63)(56,64);(1,4)(2,11)(3,17)(5,28)(6,24)(7,32)(8,25)(9,34)(10,39)(12,41)(13,42)(14,40)(15,46)(16,48)(18,54)(19,52)(20,57)(21,53)(22,26)(23,30)(27,51)(29,58)(31,59)(33,43)(35,60)(36,61)(37,47)(38,49)(44,45)(50,62)(55,64)(56,63)Polycyclic presentation:
2^9|b^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72307509, #72307518, #72307560, … |
| maximal normal subgroup of | #72307509, #72307518, #72307560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406943);