Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153167 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | #33558 | C2 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,2,7,12)(3,13,16,37)(4,9,20,36)(5,21,25,10)(6,26,28,11)(8,22,30,15)(14,42,44,40)(17,49,48,41)(18,33,51,63)(19,34,27,35)(23,52,56,38)(24,39,57,53)(29,46,60,54)(31,47,58,55)(32,62,61,43)(45,64,50,59);(1,3)(2,8)(4,14)(5,22)(6,17)(7,16)(9,32)(10,37)(11,31)(12,30)(13,21)(15,25)(18,43)(19,50)(20,44)(23,54)(24,55)(26,58)(27,45)(28,48)(29,34)(33,42)(35,60)(36,61)(38,64)(39,49)(40,63)(41,53)(46,56)(47,57)(51,62)(52,59);(1,4)(2,9)(3,14)(5,23)(6,27)(7,20)(8,29)(10,38)(11,34)(12,36)(13,40)(15,46)(16,44)(17,50)(18,24)(19,28)(21,52)(22,54)(25,56)(26,35)(30,60)(31,32)(33,39)(37,42)(41,59)(43,47)(45,48)(49,64)(51,57)(53,63)(55,62)(58,61)Polycyclic presentation:
2^9|a^2=i,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dhi,f^a=fghi,g^a=gi,c^b=cf,d^b=dhi,e^b=ei,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72286041, #72286072 |
| maximal normal subgroup of | #72286041, #72286072 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60364);