Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#3109 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1463 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,16)(4,22)(5,27)(6,13)(7,14)(8,33)(9,35)(10,20)(11,25)(12,28)(15,18)(17,50)(19,40)(21,45)(23,48)(24,56)(26,57)(29,39)(30,38)(31,46)(32,49)(34,42)(36,47)(37,59)(41,53)(43,62)(44,60)(51,61)(52,63)(54,55)(58,64);(1,3,7,20)(2,10,14,16)(4,17,26,53)(5,18,29,31)(6,30,9,19)(8,28,32,21)(11,37,44,48)(12,33,45,49)(13,40,35,38)(15,27,46,39)(22,50,57,41)(23,25,59,60)(24,52,36,51)(34,58,62,54)(42,64,43,55)(47,61,56,63);(1,4)(2,11)(3,17)(5,23)(6,24)(7,26)(8,34)(9,36)(10,37)(12,41)(13,42)(14,44)(15,47)(16,48)(18,25)(19,51)(20,53)(21,54)(22,33)(27,61)(28,58)(29,59)(30,52)(31,60)(32,62)(35,43)(38,55)(39,63)(40,64)(45,50)(46,56)(49,57);(1,5,6,21,7,29,9,28)(2,12,35,39,14,45,13,27)(3,18,30,8,20,31,19,32)(4,23,24,54,26,59,36,58)(10,33,38,15,16,49,40,46)(11,41,43,63,44,50,42,61)(17,25,52,34,53,60,51,62)(22,55,47,48,57,64,56,37);(1,6,7,9)(2,13,14,35)(3,19,20,30)(4,24,26,36)(5,28,29,21)(8,31,32,18)(10,38,16,40)(11,42,44,43)(12,39,45,27)(15,33,46,49)(17,51,53,52)(22,56,57,47)(23,58,59,54)(25,34,60,62)(37,55,48,64)(41,63,50,61)Polycyclic presentation:
2^9|b^2=i,d^2=fh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,e^b=ei,g^b=gi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530352);