Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | #3203 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1458 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)(3,12)(4,10)(5,11)(6,19)(7,9)(8,14)(13,27)(15,23)(16,25)(17,24)(18,26)(20,21)(22,29)(28,31)(30,32)(33,34,37,42,38,41,48,54)(35,39,45,50,44,51,57,61)(36,40,47,53,46,52,58,62)(43,49,56,60,55,59,63,64);(1,3,7,15)(2,8,9,21)(4,13,17,30)(5,14,6,20)(10,22,24,31)(11,23,19,12)(16,28,18,29)(25,27,26,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,49)(41,51)(42,50)(46,55)(47,56)(48,57)(52,59)(53,60)(54,61)(58,63)(62,64);(2,9)(8,21)(10,24)(11,19)(12,23)(22,31)(25,26)(27,32)(33,36,37,47,38,46,48,58)(34,40,42,53,41,52,54,62)(35,43,45,56,44,55,57,63)(39,49,50,60,51,59,61,64);(1,4)(2,10)(3,13)(5,18)(6,16)(7,17)(8,22)(9,24)(11,26)(12,27)(14,29)(15,30)(19,25)(20,28)(21,31)(23,32)(36,46)(40,52)(43,55)(47,58)(49,59)(53,62)(56,63)(60,64);(1,5,7,6)(2,11,9,19)(3,14,15,20)(4,16,17,18)(8,23,21,12)(10,25,24,26)(13,28,30,29)(22,27,31,32)(34,41)(35,44)(40,52)(42,54)(43,55)(45,57)(53,62)(56,63)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252879);