Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393907 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3835 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 3835 conjugacy classes of subgroups — too many to draw. Only its 296 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:
(1,2,8,14)(3,15,20,32)(4,21,26,36)(5,11,28,39)(6,13,7,12)(9,19,33,18)(10,25,37,24)(16,42,45,30)(17,46,49,41)(22,34,44,57)(23,52,54,50)(27,40,29,38)(31,48,58,47)(35,55,60,53)(43,61,63,56)(51,59,62,64);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,34)(11,31)(12,32)(13,15)(14,33)(16,25)(21,42)(23,51)(24,45)(26,44)(27,48)(28,49)(29,47)(30,36)(35,59)(37,57)(38,41)(39,58)(40,46)(43,53)(50,61)(52,56)(54,62)(55,63)(60,64);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,30)(11,35)(12,36)(13,21)(14,37)(15,34)(17,43)(18,22)(19,44)(20,45)(27,55)(28,54)(29,53)(31,56)(32,57)(33,42)(38,50)(39,60)(40,52)(41,59)(46,64)(47,62)(48,51)(49,63)(58,61);(1,5)(2,11)(3,17)(4,23)(6,29)(7,27)(8,28)(9,31)(10,35)(12,40)(13,38)(14,39)(15,41)(16,43)(18,48)(19,47)(20,49)(21,50)(22,51)(24,55)(25,53)(26,54)(30,56)(32,46)(33,58)(34,59)(36,52)(37,60)(42,61)(44,62)(45,63)(57,64)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72311147, #72311274, #72341344, … |
| maximal normal subgroup of | #72311147, #72311274, #72341344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301104);