Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500650 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24196 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13340 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2961 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,18,42)(4,19,22,46)(5,26,6,31)(7,36,29,13)(10,24,45,23)(11,35,12,30)(15,43,55,64)(16,50,34,48)(17,53,37,41)(20,39,60,58)(21,61,56,59)(25,51,38,49)(27,52,32,40)(28,54,33,47)(44,62,63,57);(1,3)(2,9)(4,20)(5,27)(6,32)(7,37)(8,18)(10,43)(11,48)(12,50)(13,53)(14,42)(15,24)(16,35)(17,29)(19,58)(21,33)(22,60)(23,55)(25,62)(26,52)(28,56)(30,34)(31,40)(36,41)(38,57)(39,46)(44,51)(45,64)(47,59)(49,63)(54,61);(1,4)(2,10)(3,15)(5,28)(6,33)(7,38)(8,22)(9,39)(11,49)(12,51)(13,54)(14,45)(16,57)(17,21)(18,55)(19,26)(20,32)(23,30)(24,35)(25,29)(27,60)(31,46)(34,62)(36,47)(37,56)(40,61)(41,44)(42,58)(43,50)(48,64)(52,59)(53,63);(1,3)(2,9)(4,21,22,56)(5,16)(6,34)(7,17)(8,18)(10,44,45,63)(11,40)(12,52)(13,41)(14,42)(15,38,55,25)(19,59,46,61)(20,33,60,28)(23,62,24,57)(26,50)(27,35)(29,37)(30,32)(31,48)(36,53)(39,54,58,47)(43,51,64,49)Polycyclic presentation:
2^9|a^2=i,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gh,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249766, #72249769, … |
| maximal normal subgroup of | #72249754, #72249766, #72249769, … |
| maximal quotient of | #72249886, #72249997, #72250002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407847);