Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13407 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33720 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 64 points:
(1,2)(3,17)(4,10)(5,25)(6,31)(7,35)(8,12)(9,16)(11,20)(13,26)(14,34)(15,28)(18,38)(19,42)(21,60)(22,40)(23,47)(24,51)(27,63)(29,46)(30,49)(32,39)(33,43)(36,54)(37,53)(41,58)(44,55)(45,62)(48,57)(50,56)(52,59)(61,64);(2,10)(3,18)(4,8)(5,26)(6,32)(7,28)(11,40)(12,16)(13,47)(14,49)(15,51)(17,44)(19,20)(21,27)(22,58)(23,34)(24,29)(25,39)(30,62)(31,45)(33,36)(37,55)(38,42)(41,53)(43,50)(48,52)(54,63)(56,59)(57,61)(60,64);(1,3,4,19,9,22,8,37)(2,11,12,41,16,44,10,38)(5,27,23,61,30,59,6,33)(7,36,24,56,29,57,28,21)(13,48,14,50,45,54,39,60)(15,52,46,64,51,63,35,43)(17,47,53,25,40,31,42,49)(18,26,55,32,58,62,20,34)Polycyclic presentation:
2^9|c^2=d,d^2=i,e^2=gh,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=eg,f^b=fgi,g^b=gh,h^b=hi,e^c=egi,f^c=fg,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72224988, #72225000, #72225007, … |
| maximal normal subgroup of | #72224988, #72225000, #72225007, … |
| maximal quotient of | C2×#153274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60470);