Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153607 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13661 |
| commutator, Frattini, cosocle | C2×C32 | C23 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C22×C16 ◁ C2×M64 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(2,8)(3,13)(4,16)(5,20)(6,24)(9,27)(10,32)(11,34)(12,35)(14,19)(15,36)(17,47)(18,49)(21,55)(22,42)(23,28)(25,54)(26,29)(30,59)(31,60)(33,61)(37,38)(39,44)(40,51)(41,56)(43,64)(45,46)(48,52)(50,53)(57,58)(62,63);(1,2)(3,14)(4,19)(5,23)(6,26)(7,9)(8,13)(10,16)(11,20)(12,24)(15,40)(17,48)(18,50)(21,56)(22,51)(25,57)(27,36)(28,47)(29,49)(30,42)(31,54)(32,37)(33,55)(34,38)(35,39)(41,44)(43,61)(45,53)(46,52)(58,64)(59,62)(60,63);(3,15)(8,27)(13,36)(14,40)(17,43)(18,44)(28,58)(29,35)(32,59)(34,60)(37,62)(38,63)(39,49)(41,50)(47,64)(48,61)Polycyclic presentation:
2^9|d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,e^b=ei,f^b=fg,g^b=gh,h^b=hi,d^c=di53 conjugacy classes — view the full character table.
| maximal subgroup of | #72225424, #72321773, #72321786, … |
| maximal normal subgroup of | #72225424, #72321773, #72321786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60803);