Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60972 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,9,13,23)(3,15,17,46)(4,20)(5,22)(6,28)(10,36,40,63)(11,34,12,32)(14,35,33,56)(16,48,24,58)(18,30,19,47)(21,54)(26,42)(27,50)(29,59,44,61)(31,60,49,37)(38,43,57,39)(41,64,53,62)(45,52,55,51);(1,2,5,23,25,33,50,35)(3,10,24,57,17,53,16,49)(4,11,21,34,28,44,54,61)(6,29,26,59,20,12,42,32)(7,14,22,56,8,13,27,9)(15,37,55,63,46,39,45,64)(18,40,52,38,19,41,51,31)(30,60,58,36,47,43,48,62);(1,3)(2,10)(4,15)(5,24)(6,30)(7,18)(8,19)(9,31)(11,37)(12,43)(13,41)(14,40)(16,50)(17,25)(20,47)(21,55)(22,52)(23,57)(26,58)(27,51)(28,46)(29,60)(32,62)(33,53)(34,63)(35,49)(36,59)(38,56)(39,44)(42,48)(45,54)(61,64);(1,4,7,6)(2,11,14,29)(3,15,18,30)(5,21,22,26)(8,20,25,28)(9,32,35,61)(10,37,40,60)(12,33,44,13)(16,45,51,48)(17,46,19,47)(23,34,56,59)(24,55,52,58)(27,42,50,54)(31,62,49,64)(36,57,63,38)(39,41,43,53)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72245124 |
| maximal normal subgroup of | #72245124 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107928);