Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29636 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #13350 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
803 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 803 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,13)(4,19)(5,10)(6,26)(7,12)(8,16)(9,22)(11,29)(14,45)(15,43)(17,44)(18,40)(20,36)(21,35)(23,56)(24,59)(25,39)(27,33)(28,55)(30,61)(31,51)(32,53)(34,46)(37,58)(38,60)(41,57)(42,48)(47,54)(49,62)(50,63)(52,64);(1,3)(2,8)(4,14)(5,15)(6,27)(7,18)(9,34)(10,31)(11,40)(12,33)(13,26)(16,29)(17,25)(19,54)(20,46)(21,52)(22,48)(23,49)(24,53)(28,47)(30,37)(32,39)(35,45)(36,63)(38,44)(41,42)(43,59)(50,58)(51,60)(55,62)(56,64)(57,61);(1,4)(2,9)(3,14)(5,22)(6,28)(7,23)(8,30)(10,19)(11,41)(12,37)(13,42)(15,50)(16,52)(17,48)(18,49)(20,24)(21,29)(25,58)(26,36)(27,47)(31,54)(32,64)(33,34)(35,38)(39,56)(40,63)(43,62)(44,45)(46,51)(53,61)(55,59)(57,60);(1,5,7,25)(2,10,12,39)(3,15,18,17)(4,20)(6,24,29,60)(8,31,33,32)(9,35)(11,38,26,59)(13,43,40,44)(14,46)(16,51,27,53)(19,36)(21,22)(23,57)(28,58)(30,62)(34,45)(37,55)(41,56)(42,64)(47,50)(48,52)(49,61)(54,63)Polycyclic presentation:
2^9|d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gh,c^b=cg,e^b=ei,f^b=fh,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415300);