Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8375 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61081 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 793 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,7,14,9,15,33,50,8,16,34,41,6,13,24,48)(3,17,21,53,20,44,59,37,22,54,60,63,35,62,32,61)(4,23,26,45,25,43,5,12,27,46,30,47,31,11,29,42)(10,36,39,58,38,18,49,55,40,57,52,19,51,56,64,28);(1,3)(2,10)(4,18)(5,28)(6,20)(7,32)(8,22)(9,35)(11,37)(12,44)(13,38)(14,49)(15,51)(16,40)(17,47)(19,30)(21,24)(23,53)(25,36)(26,58)(27,56)(29,55)(31,57)(33,60)(34,59)(39,50)(41,64)(42,62)(43,61)(45,54)(46,63)(48,52);(1,4,8,27)(2,11,16,43)(3,18,22,56)(5,24,29,33)(6,31,9,25)(7,26,34,30)(10,37,40,61)(12,41,42,14)(13,46,15,23)(17,52,54,39)(19,32,58,59)(20,57,35,36)(21,55,60,28)(38,63,51,53)(44,64,62,49)(45,50,47,48)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72307133, #72307135, #72307140, … |
| maximal normal subgroup of | #72307133, #72307135, #72307140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37977);