Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60942 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1528 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,15)(14,27)(16,22)(17,26)(18,24)(19,25)(20,23)(21,29)(28,32)(30,31)(33,34,37,42,38,41,48,54)(35,39,45,51,44,50,57,61)(36,40,47,53,46,52,58,62)(43,49,55,60,56,59,63,64);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,22)(12,23)(17,28)(18,29)(19,30)(24,31)(25,32)(26,27)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,50)(42,51)(46,56)(47,55)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(1,4)(2,9)(3,14)(5,18)(6,17)(7,19)(8,21)(10,26)(11,24)(12,25)(13,27)(15,29)(16,30)(20,28)(22,31)(23,32)(33,36,37,47,38,46,48,58)(34,40,42,53,41,52,54,62)(35,43,45,55,44,56,57,63)(39,49,51,60,50,59,61,64);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,17,19,18)(8,13,23,22)(9,24,25,26)(14,28,30,29)(21,31,32,27)(35,44)(36,46)(39,50)(40,52)(45,57)(47,58)(51,61)(53,62);(1,5,7,6)(2,11,12,10)(3,15,16,20)(4,18,19,17)(8,22,23,13)(9,24,25,26)(14,29,30,28)(21,31,32,27)(34,41)(39,50)(40,52)(42,54)(49,59)(51,61)(53,62)(60,64)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,d^b=di,f^b=fh,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252682);