Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186976 |
| 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 | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2002 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)(4,15,17,8)(5,9)(6,10)(12,20)(13,23,18,21)(16,28,30,22)(19,24)(25,27)(26,31)(29,32)(33,34)(35,43,46,57)(36,40)(37,42)(38,41)(39,45,53,49)(44,60,62,59)(47,55)(48,54)(50,58)(51,56)(52,61,64,63);(1,3,6,14)(2,7,10,11)(4,12,17,27)(5,18,19,13)(8,20,15,25)(9,23,24,21)(16,29,30,26)(22,32,28,31)(33,35)(34,39)(36,44)(37,49)(38,46)(40,52)(41,53)(42,57)(43,56)(45,51)(47,63)(48,62)(50,61)(54,64)(55,59)(58,60);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,25)(13,26)(14,27)(18,29)(19,30)(21,31)(23,32)(24,28)(33,36,38,48)(34,40,41,54)(35,44,46,62)(37,50,51,47)(39,52,53,64)(42,58,56,55)(43,59,57,60)(45,63,49,61);(1,5,6,19)(2,9,10,24)(3,13,14,18)(4,16,17,30)(7,21,11,23)(8,22,15,28)(12,26,27,29)(20,31,25,32)(34,41)(39,53)(40,54)(42,56)(43,57)(52,64)(55,58)(59,60)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,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, 94213);