Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3184 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23573 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
802 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 4 generators on 64 points:
(1,2)(3,8)(4,17)(5,22)(6,27)(7,11)(9,20)(10,26)(12,25)(13,38)(14,29)(15,44)(16,47)(18,36)(19,50)(21,53)(23,33)(24,54)(28,41)(30,46)(31,43)(32,60)(34,52)(35,51)(37,55)(39,61)(40,49)(42,63)(45,59)(48,58)(56,62)(57,64);(1,3)(2,8)(4,13)(5,15)(6,16)(7,14)(9,32)(10,30)(11,29)(12,31)(17,49)(18,48)(19,40)(20,41)(21,39)(22,44)(23,45)(24,42)(25,43)(26,46)(27,47)(28,34)(33,56)(35,59)(36,61)(37,57)(38,50)(51,62)(52,60)(53,58)(54,63)(55,64);(1,4)(2,9)(3,13)(5,23)(6,18)(7,19)(8,28)(10,36)(11,34)(12,33)(14,40)(15,45)(16,48)(17,22)(20,26)(21,25)(24,51)(27,35)(29,32)(30,58)(31,59)(37,53)(38,44)(39,43)(41,46)(42,62)(47,56)(49,63)(50,54)(52,55)(57,61)(60,64);(3,14)(4,18,19,21)(5,24)(8,29)(9,33,34,35)(10,37)(13,39,40,48)(16,43)(17,36,50,53)(20,23,52,51)(22,54)(26,55)(28,56,32,59)(31,47)(38,61,49,58)(41,62,60,45)Polycyclic presentation:
2^9|d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=dh,e^c=ei,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72249759, #72249766, #72249835, … |
| maximal normal subgroup of | #72249759, #72249766, #72249835, … |
| maximal quotient of | #72250129, #72250218, #72250299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406458);