Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2044 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#2044 |
| prime radical | #60749 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1088 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)(3,14)(4,19)(5,11)(6,26)(7,29)(8,13)(9,16)(10,23)(12,31)(15,43)(17,47)(18,33)(20,51)(21,49)(22,55)(24,50)(25,58)(27,35)(28,39)(30,42)(32,61)(34,59)(36,64)(37,54)(38,57)(40,46)(41,45)(44,63)(48,52)(53,62)(56,60);(1,3)(2,9)(4,20)(5,16)(6,27)(7,30)(8,18)(10,36)(11,33)(12,40)(13,35)(14,26)(15,24)(17,31)(19,48)(21,52)(22,44)(23,53)(25,59)(28,46)(29,34)(32,38)(37,41)(39,42)(43,49)(45,57)(47,58)(50,63)(51,55)(54,64)(56,61)(60,62);(1,4,6,22,8,24,5,21)(2,10,11,37,13,38,26,60)(3,15,16,44,18,20,27,52)(7,23,28,56,31,57,25,54)(9,32,14,41,35,36,33,62)(12,19,39,49,29,50,58,55)(17,45,46,61,30,53,59,64)(34,63,42,43,40,48,47,51);(1,5,8,6)(2,11,13,26)(3,16,18,27)(4,21,24,22)(7,25,31,28)(9,33,35,14)(10,37,38,60)(12,39,29,58)(15,44,20,52)(17,46,30,59)(19,49,50,55)(23,54,57,56)(32,62,36,41)(34,47,40,42)(43,63,51,48)(45,61,53,64)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,g^c=gi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303731, #72303740, #72303747, … |
| maximal normal subgroup of | #72303731, #72303740, #72303747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406346);