Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500797 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1020 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)(3,15)(4,20)(5,9)(6,27)(7,13)(8,14)(10,23)(11,19)(12,16)(17,45)(18,42)(21,48)(22,54)(24,51)(25,32)(26,33)(28,59)(29,37)(30,40)(31,52)(34,60)(35,62)(36,56)(38,44)(39,57)(41,46)(43,50)(47,49)(53,63)(55,58)(61,64);(1,3,16,26,8,19,6,5)(2,9,27,11,14,33,12,15)(4,17,43,53,24,47,22,21)(7,29,44,25,30,18,28,57)(10,31,58,34,36,61,35,41)(13,39,59,42,40,32,38,37)(20,48,54,49,51,63,50,45)(23,46,62,64,56,60,55,52);(1,4)(2,10)(3,17)(5,21)(6,22)(7,23)(8,24)(9,31)(11,34)(12,35)(13,20)(14,36)(15,41)(16,43)(18,46)(19,47)(25,52)(26,53)(27,58)(28,55)(29,60)(30,56)(32,48)(33,61)(37,49)(38,50)(39,63)(40,51)(42,45)(44,62)(54,59)(57,64);(1,5,6,19,8,26,16,3)(2,11,12,9,14,15,27,33)(4,21,22,47,24,53,43,17)(7,25,28,29,30,57,44,18)(10,34,35,31,36,41,58,61)(13,37,38,32,40,42,59,39)(20,49,50,48,51,45,54,63)(23,52,55,60,56,64,62,46)Polycyclic presentation:
2^9|b^2=ei,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312437, #72312458, #72312501, … |
| maximal normal subgroup of | #72312437, #72312458, #72312501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407994);