Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187168 |
| 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)(3,13,17,23,16,8,15,24)(4,18,19,26,20,9,21,25)(14,27,28,32,29,22,30,31)(33,34)(35,44)(36,49,52,41)(37,42)(38,53)(39,43)(40,47)(45,59,62,58)(46,55)(48,57)(50,64,51,56)(54,61,63,60);(1,3,7,16)(2,8,11,13)(4,14,20,29)(5,17,6,15)(9,22,18,27)(10,24,12,23)(19,30,21,28)(25,32,26,31)(33,35,39,48)(34,40,43,55)(36,45,52,62)(37,47,38,46)(41,54,49,63)(42,57,53,44)(50,61,51,60)(56,59,64,58);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,28)(16,29)(17,30)(23,31)(24,32)(33,36)(34,41)(35,45)(37,50)(38,51)(39,52)(40,54)(42,56)(43,49)(44,58)(46,60)(47,61)(48,62)(53,64)(55,63)(57,59);(1,5,7,6)(2,10,11,12)(3,15,16,17)(4,19,20,21)(8,23,13,24)(9,25,18,26)(14,28,29,30)(22,31,27,32)(33,37,39,38)(34,42,43,53)(35,46,48,47)(36,50,52,51)(40,44,55,57)(41,56,49,64)(45,60,62,61)(54,58,63,59)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=fhi,g^b=ghi,e^c=ehi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94552);