Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60400 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1825 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1825 conjugacy classes of subgroups — too many to draw. Only its 148 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)(4,9)(5,10)(6,25)(7,12)(8,15)(11,28)(14,43)(16,41)(17,49)(18,51)(19,33)(20,39)(21,35)(22,54)(23,36)(24,58)(26,42)(27,34)(29,46)(30,59)(31,48)(32,52)(37,57)(38,47)(40,45)(44,60)(50,64)(53,63)(55,61)(56,62);(1,3,7,16)(2,8,12,31)(4,18,21,50)(5,15,23,48)(6,26,28,17)(9,32,35,61)(10,13,36,41)(11,38,25,30)(14,27,45,20)(19,52,22,55)(24,59,57,47)(29,39,60,34)(33,51,54,64)(37,49,58,42)(40,63,43,62)(44,56,46,53);(1,4,7,21)(2,9,12,35)(3,14,16,45)(5,22,23,19)(6,27,28,20)(8,29,31,60)(10,33,36,54)(11,39,25,34)(13,40,41,43)(15,46,48,44)(17,50,26,18)(24,53,57,56)(30,61,38,32)(37,63,58,62)(42,64,49,51)(47,52,59,55);(2,10)(8,13)(9,33)(11,37)(12,36)(25,58)(29,40)(30,42)(31,41)(32,51)(34,62)(35,54)(38,49)(39,63)(43,60)(61,64)Polycyclic presentation:
2^9|b^2=i,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72281917, #72282105, #72282134, … |
| maximal normal subgroup of | #72281917, #72282105, #72282134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407018);