Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187949 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62823 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2008 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)(3,11,14,7)(4,8)(5,17)(6,10)(9,19)(12,25,28,20)(13,21,30,26)(15,24)(16,23)(18,22)(27,31,29,32)(33,34,37,42,38,41,48,54)(35,39,45,51,44,50,58,62)(36,40,47,53,46,52,59,63)(43,55,57,64,56,49,60,61);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,21)(10,11)(15,27)(16,28)(17,26)(18,29)(19,30)(22,31)(23,25)(24,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,55)(53,61)(54,62)(59,60)(63,64);(1,4,6,16)(2,8,10,23)(3,12,14,28)(5,18,19,15)(7,20,11,25)(9,24,17,22)(13,29,30,27)(21,32,26,31)(33,36)(34,40)(35,43)(37,47)(38,46)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,55)(51,61)(54,63)(58,60)(62,64);(1,5,6,19)(2,9,10,17)(3,13,14,30)(4,15,16,18)(7,21,11,26)(8,22,23,24)(12,27,28,29)(20,31,25,32)(34,41)(39,50)(40,52)(42,54)(49,55)(51,62)(53,63)(61,64)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95221);