Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #300501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62558 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
858 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)(4,15)(5,19)(6,10)(7,13)(8,17)(9,20)(12,26)(14,25)(16,29)(18,30)(21,28)(22,23)(24,27)(31,32)(33,34)(35,44)(36,48)(37,42)(38,52)(39,43)(40,46)(41,51)(45,62)(47,56)(49,55)(50,60)(53,57)(54,59)(58,63)(61,64);(1,3,5,13,6,14,20,23)(2,7,9,11,10,22,19,25)(4,12,17,27,18,28,29,31)(8,21,15,24,16,26,30,32)(33,35,37,46)(34,40,42,44)(36,49,50,64)(38,47,39,53)(41,58,59,62)(43,56,52,57)(45,54,63,51)(48,61,60,55);(1,2)(3,7,14,22)(4,16)(5,9)(6,10)(8,18)(11,23,25,13)(12,26,28,21)(15,29)(17,30)(19,20)(24,27,32,31)(33,36,38,51,37,50,39,54)(34,41,43,48,42,59,52,60)(35,45,47,64,46,63,53,49)(40,55,56,58,44,61,57,62);(1,4,5,17,6,18,20,29)(2,8,9,15,10,16,19,30)(3,12,13,27,14,28,23,31)(7,21,11,24,22,26,25,32)(34,42)(36,50)(40,44)(43,52)(45,63)(49,64)(51,54)(56,57)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 207713);