Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352469 |
| 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 | #62872 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2630 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)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(33,34,38,42)(35,39,45,51)(36,40,47,53)(37,41,48,54)(43,55,57,49)(44,50,58,61)(46,52,59,62)(56,63,64,60);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,27)(25,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,60)(53,55)(54,61)(59,64)(62,63);(1,4,7,17)(2,9,12,23)(3,14,16,28)(5,18,6,19)(8,21,22,31)(10,24,11,25)(13,27,26,32)(15,29,20,30)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,60)(51,55)(54,62)(58,64)(61,63);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(37,48)(41,54)(44,58)(46,59)(50,61)(52,62)(56,64)(60,63);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,18,17,19)(8,13,22,26)(9,24,23,25)(14,29,28,30)(21,27,31,32)(33,37,38,48)(34,41,42,54)(35,44,45,58)(36,46,47,59)(39,50,51,61)(40,52,53,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259766);