Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,19)(14,28)(16,24)(17,23)(20,25)(21,26)(22,30)(27,29)(31,32)(33,34,38,44,39,42,37,43)(35,45,49,40,47,55,48,56)(36,50,52,41,53,58,51,57)(46,54,62,59,60,64,61,63);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,19,29,20,30,21,31)(9,22,18,27,26,28,25,32)(33,35)(34,40)(36,46)(37,49)(38,48)(39,47)(41,54)(42,56)(43,55)(44,45)(50,59)(51,62)(52,61)(53,60)(57,64)(58,63);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)(33,36,39,53)(34,41,42,57)(35,46,47,60)(37,52,38,51)(40,54,56,64)(43,58,44,50)(45,59,55,63)(48,62,49,61);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,25,26,18)(14,29,30,31)(22,32,28,27)(34,42)(35,47)(41,57)(43,44)(46,60)(48,49)(50,58)(61,62)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105880);