Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186976 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62831 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,5,10)(3,13,16,23,17,8,15,24)(4,18,19,26,20,9,21,25)(14,22,28,31,29,27,30,32)(33,34)(35,44)(36,49,51,41)(37,42)(38,52)(39,43)(40,47)(45,59,61,64)(46,55)(48,58)(50,57,53,56)(54,60,63,62);(1,3,7,17)(2,8,11,13)(4,14,20,29)(5,16,6,15)(9,22,18,27)(10,24,12,23)(19,30,21,28)(25,32,26,31)(33,35,39,48)(34,40,43,55)(36,45,51,61)(37,47,38,46)(41,54,49,63)(42,58,52,44)(50,62,53,60)(56,64,57,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,28)(16,30)(17,29)(23,31)(24,32)(33,36)(34,41)(35,45)(37,50)(38,53)(39,51)(40,54)(42,56)(43,49)(44,59)(46,60)(47,62)(48,61)(52,57)(55,63)(58,64);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,20,21)(8,23,13,24)(9,25,18,26)(14,28,29,30)(22,31,27,32)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,50,51,53)(40,44,55,58)(41,56,49,57)(45,60,61,62)(54,59,63,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94320);