Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512866 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #127 | C24 |
| prime radical | #61111 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1015 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,7,12)(3,8,16,30)(4,9,20,34)(5,21,24,10)(6,25,26,11)(13,37,41,54)(14,35,22,28)(15,29,43,52)(17,44,48,31)(18,32,19,33)(23,36,51,49)(27,40,55,39)(38,62,45,61)(42,60,50,56)(46,59,47,58)(53,63,57,64);(1,3,7,16)(2,8,12,30)(4,13,20,41)(5,22,24,14)(6,15,26,43)(9,27,34,55)(10,35,21,28)(11,29,25,52)(17,45,48,38)(18,39,19,40)(23,50,51,42)(31,57,44,53)(32,37,33,54)(36,60,49,56)(46,64,47,63)(58,62,59,61);(1,4)(2,9)(3,13)(5,17)(6,18)(7,20)(8,27)(10,31)(11,32)(12,34)(14,38)(15,39)(16,41)(19,26)(21,44)(22,45)(23,46)(24,48)(25,33)(28,53)(29,37)(30,55)(35,57)(36,58)(40,43)(42,63)(47,51)(49,59)(50,64)(52,54)(56,61)(60,62);(1,5,7,24)(2,10,12,21)(3,14,16,22)(4,17,20,48)(6,23,26,51)(8,28,30,35)(9,31,34,44)(11,36,25,49)(13,38,41,45)(15,42,43,50)(18,46,19,47)(27,53,55,57)(29,56,52,60)(32,58,33,59)(37,61,54,62)(39,63,40,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,d^b=di,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72306615, #72306631, #72306632, … |
| maximal normal subgroup of | #72306615, #72306631, #72306632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420066);