Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187195 |
| 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 | #62752 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,19)(3,12,14,23,15,8,20,24)(4,16,17,25,18,9,21,26)(13,27,30,32,29,22,28,31)(33,34)(35,43)(36,47,49,40)(37,50)(38,42)(39,45)(41,52)(44,58,60,57)(46,56)(48,55,64,62)(51,54)(53,59,61,63);(1,3,7,15)(2,8,11,12)(4,13,18,29)(5,14,6,20)(9,22,16,27)(10,23,19,24)(17,28,21,30)(25,31,26,32)(33,35,38,46)(34,39,42,54)(36,44,49,60)(37,51,52,45)(40,53,47,61)(41,56,50,43)(48,63,64,59)(55,58,62,57);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,16)(12,27)(14,28)(15,29)(19,26)(20,30)(23,31)(24,32)(33,36)(34,40)(35,44)(37,48)(38,49)(39,53)(41,55)(42,47)(43,57)(45,59)(46,60)(50,62)(51,63)(52,64)(54,61)(56,58);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,18,21)(8,23,12,24)(9,25,16,26)(13,28,29,30)(22,31,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94573);