Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199534 |
| 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 | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
976 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)(4,18,21,25)(5,10)(6,11)(7,12)(8,16)(9,20,26,19)(14,28,31,22)(15,24)(17,23)(27,30,32,29)(33,34)(35,39,45,54)(36,46,48,40)(37,41)(38,42)(43,52)(44,53,49,56)(47,57,50,55)(51,58)(59,63)(60,62)(61,64);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,23)(4,14,20,29,21,31,19,30)(9,22,25,27,26,28,18,32)(33,35)(34,39,42,54)(36,43)(37,49)(38,45)(40,52,46,62)(41,56,58,53)(44,51)(47,61)(48,60)(50,59)(55,64,57,63);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,25)(12,26)(13,27)(15,30)(16,29)(17,31)(23,32)(24,28)(33,36)(34,40)(35,43)(37,50)(38,48)(39,52)(41,57)(42,46)(44,61)(45,60)(47,51)(49,59)(53,64)(54,62)(55,58)(56,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,25,26,18)(14,29,31,30)(22,32,28,27)(33,37,38,51)(34,41,42,58)(35,44,45,49)(36,47,48,50)(39,53,54,56)(40,55,46,57)(43,59,60,61)(52,63,62,64)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,d^b=dh,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, 106731);