Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347544 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,22)(4,9,18,24)(5,11,6,10)(8,15,23,16)(14,27,28,32)(19,26,20,25)(21,29,31,30)(33,34)(35,40)(36,41)(37,51)(38,42)(39,43)(44,57)(45,61)(46,53)(47,54)(48,62)(49,55)(50,56)(52,60)(58,63)(59,64);(1,3,7,17)(2,8,12,23)(4,14,18,28)(5,16,6,15)(9,21,24,31)(10,13,11,22)(19,30,20,29)(25,27,26,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,52)(42,53)(43,54)(48,58)(49,59)(50,60)(51,61)(55,63)(56,57)(62,64);(1,4,7,18)(2,9,12,24)(3,14,17,28)(5,19,6,20)(8,21,23,31)(10,25,11,26)(13,27,22,32)(15,29,16,30)(33,36)(34,41)(35,44)(37,48)(38,49)(39,50)(40,52)(42,55)(43,56)(45,58)(46,59)(47,60)(51,62)(53,63)(54,57)(61,64);(4,18)(9,24)(14,28)(19,20)(21,31)(25,26)(27,32)(29,30)(33,37,39,38)(34,42,43,51)(35,45,47,46)(36,48,50,49)(40,53,54,61)(41,55,56,62)(44,58,60,59)(52,63,57,64);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,18,20)(8,22,23,13)(9,25,24,26)(14,29,28,30)(21,32,31,27)(33,38,39,37)(34,42,43,51)(35,46,47,45)(36,49,50,48)(40,53,54,61)(41,55,56,62)(44,59,60,58)(52,63,57,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254790);