Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130786 |
| 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 | #61090 | 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 | |
807 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 807 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:
(1,2,7,14,8,13,30,48,9,15,36,42,6,33,24,50)(3,16,20,52,21,45,35,63,22,53,61,64,19,60,57,38)(4,23,26,46,25,11,5,28,27,44,31,62,34,43,32,12)(10,37,40,18,39,17,54,29,41,55,51,59,47,56,49,58);(1,3)(2,10)(4,17)(5,29)(6,21)(7,35)(8,19)(9,22)(11,38)(12,45)(13,47)(14,49)(15,41)(16,46)(18,31)(20,30)(23,52)(24,61)(25,37)(26,59)(27,56)(28,60)(32,58)(33,39)(34,55)(36,57)(40,50)(42,54)(43,63)(44,64)(48,51)(53,62);(1,4,9,27)(2,11,15,43)(3,17,22,56)(5,30,32,24)(6,34,8,25)(7,31,36,26)(10,38,41,63)(12,14,28,42)(13,44,33,23)(16,51,53,40)(18,57,59,35)(19,37,21,55)(20,58,61,29)(39,52,47,64)(45,49,60,54)(46,48,62,50)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,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72307224, #72307293, #72307560, … |
| maximal normal subgroup of | #72307224, #72307293, #72307560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37983);