Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199528 |
| 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 | C23×C4 | C24 |
| prime radical | #60970 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
984 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)(4,17,20,25)(5,10)(6,11)(7,12)(8,15)(9,19,26,18)(14,28,30,22)(16,23)(21,24)(27,29,32,31)(33,34)(35,39,45,53)(36,46,48,40)(37,41)(38,42)(43,51)(44,52,60,64)(47,55,49,54)(50,56)(57,62)(58,61)(59,63);(1,3,5,15,7,16,6,21)(2,8,11,13,12,24,10,23)(4,14,19,31,20,30,18,29)(9,22,25,27,26,28,17,32)(33,35)(34,39,42,53)(36,43)(37,44)(38,45)(40,51,46,61)(41,52,56,64)(47,57)(48,58)(49,59)(50,60)(54,62,55,63);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,22)(10,17)(11,25)(12,26)(13,27)(15,31)(16,30)(21,29)(23,32)(24,28)(33,36)(34,40)(35,43)(37,49)(38,48)(39,51)(41,55)(42,46)(44,59)(45,58)(47,50)(52,63)(53,61)(54,56)(57,60)(62,64);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,25,26,17)(14,29,30,31)(22,27,28,32)(33,37,38,50)(34,41,42,56)(35,44,45,60)(36,47,48,49)(39,52,53,64)(40,54,46,55)(43,57,58,59)(51,62,61,63)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106726);