Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501114 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #23874 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1059 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15)(4,10)(5,25)(6,12,30,43)(7,13)(8,14)(9,18,34,27)(11,26,42,29)(16,47)(17,33)(19,44)(20,54)(21,46)(22,38,28,59)(23,39)(24,40)(31,60)(32,48,61,52)(35,51,45,53)(36,55,63,57)(37,56,41,58)(49,64)(50,62);(1,3,8,19)(2,9,14,34)(4,20,24,50)(5,17,7,31)(6,27,30,18)(10,36,40,63)(11,33,42,60)(12,44,43,15)(13,45,25,35)(16,21,49,23)(22,57,28,55)(26,53,29,51)(32,46,61,39)(37,64,41,47)(38,62,59,54)(48,58,52,56);(1,4,8,24)(2,10,14,40)(3,16,19,49)(5,23,7,21)(6,28,30,22)(9,32,34,61)(11,41,42,37)(12,38,43,59)(13,46,25,39)(15,47,44,64)(17,50,31,20)(18,52,27,48)(26,56,29,58)(33,62,60,54)(35,63,45,36)(51,55,53,57);(1,5,8,7)(2,11,14,42)(3,17,19,31)(4,21,24,23)(6,29,30,26)(9,33,34,60)(10,37,40,41)(12,13,43,25)(15,35,44,45)(16,20,49,50)(18,53,27,51)(22,58,28,56)(32,54,61,62)(36,64,63,47)(38,39,59,46)(48,57,52,55)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72312437, #72312488, #72312496, … |
| maximal normal subgroup of | #72312437, #72312488, #72312496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408315);