Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357716 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3511 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,11)(3,12,17,28)(4,9,18,8)(5,20,13,10)(7,21,23,16)(14,30,15,27)(19,25,29,26)(22,31,24,32)(33,34,38,42)(35,39,44,52)(36,46,49,61)(37,50,47,41)(40,54,55,48)(43,56,58,64)(45,60,57,53)(51,62,63,59);(1,3,13,21,6,17,5,16)(2,7,20,28,11,23,10,12)(4,15,29,31,18,14,19,32)(8,24,26,27,9,22,25,30)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,53)(42,52)(46,56)(47,57)(48,59)(49,58)(50,60)(54,62)(55,63)(61,64);(1,4)(2,8)(3,14)(5,19)(6,18)(7,22)(9,11)(10,25)(12,27)(13,29)(15,17)(16,31)(20,26)(21,32)(23,24)(28,30)(33,36,47,54,38,49,37,48)(34,40,50,61,42,55,41,46)(35,43,57,62,44,58,45,59)(39,51,60,64,52,63,53,56);(1,4)(2,9)(3,15)(5,19)(6,18)(7,22)(8,11)(10,26)(12,27)(13,29)(14,17)(16,32)(20,25)(21,31)(23,24)(28,30)(35,44)(39,52)(43,58)(45,57)(51,63)(53,60)(56,64)(59,62);(4,18)(8,9)(14,15)(19,29)(22,24)(25,26)(27,30)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=ghi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264943);