Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199522 |
| 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 | #60954 | 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)(3,12)(4,16,19,26)(5,20)(6,10)(7,11)(8,15)(9,18,25,17)(13,28,29,22)(14,24)(21,23)(27,30,32,31)(33,34)(35,39,45,53)(36,46,48,40)(37,41)(38,42)(43,51)(44,52,49,55)(47,60,62,54)(50,56)(57,63)(58,59)(61,64);(1,3,5,15,7,14,6,21)(2,8,10,12,11,23,20,24)(4,13,18,31,19,29,17,30)(9,22,26,27,25,28,16,32)(33,35)(34,39,42,53)(36,43)(37,49)(38,45)(40,51,46,59)(41,55,56,52)(44,50)(47,61)(48,58)(54,64,60,63)(57,62);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,22)(10,26)(11,25)(12,27)(14,29)(15,31)(16,20)(21,30)(23,28)(24,32)(33,36)(34,40)(35,43)(37,47)(38,48)(39,51)(41,54)(42,46)(44,57)(45,58)(49,61)(50,62)(52,63)(53,59)(55,64)(56,60);(3,14)(8,23)(12,24)(13,29)(15,21)(22,28)(27,32)(30,31)(33,37,38,50)(34,41,42,56)(35,44,45,49)(36,47,48,62)(39,52,53,55)(40,54,46,60)(43,57,58,61)(51,63,59,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,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,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, 106721);