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 | |
805 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 805 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:
(1,2,7,14,6,33,30,50,9,15,36,45,8,13,24,48)(3,16,20,51,19,57,53,63,22,52,59,60,21,44,35,38)(4,23,26,46,34,42,32,12,27,43,31,61,25,11,5,28)(10,37,40,56,47,17,49,29,41,58,62,18,39,54,64,55);(1,3)(2,10)(4,17)(5,29)(6,21)(7,35)(8,19)(9,22)(11,38)(12,44)(13,47)(14,49)(15,41)(16,46)(18,31)(20,24)(23,60)(25,58)(26,56)(27,54)(28,57)(30,59)(32,55)(33,39)(34,37)(36,53)(40,50)(42,63)(43,51)(45,64)(48,62)(52,61);(1,4)(2,11)(3,17)(5,30)(6,34)(7,31)(8,25)(9,27)(10,38)(12,45)(13,43)(14,28)(15,42)(16,40)(18,35)(19,58)(20,55)(21,37)(22,54)(23,33)(24,32)(26,36)(29,59)(39,60)(41,63)(44,64)(46,50)(47,51)(48,61)(49,57)(52,62)(53,56)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,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 | #72307079, #72307103, #72307509, … |
| maximal normal subgroup of | #72307079, #72307103, #72307509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37986);