Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | C23×C4⋊C4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1482 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,15)(13,26)(14,22)(16,27)(17,23)(18,24)(19,25)(21,29)(28,32)(30,31)(33,34,37,41,38,42,48,54)(35,39,45,51,44,50,57,61)(36,40,47,53,46,52,58,62)(43,49,56,60,55,59,63,64);(1,3,6,15,7,16,5,14)(2,8,10,12,11,22,20,27)(4,13,18,29,17,28,19,30)(9,21,25,26,23,31,24,32)(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);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,21)(10,25)(11,23)(12,26)(14,30)(15,29)(16,28)(20,24)(22,31)(27,32)(33,36)(34,40)(35,43)(37,47)(38,46)(39,49)(41,53)(42,52)(44,55)(45,56)(48,58)(50,59)(51,60)(54,62)(57,63)(61,64);(4,17)(9,23)(13,28)(18,19)(21,31)(24,25)(26,32)(29,30)(36,46)(40,52)(43,55)(47,58)(49,59)(53,62)(56,63)(60,64);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,17,19)(8,12,22,27)(9,24,23,25)(13,29,28,30)(21,32,31,26)(35,44)(39,50)(43,55)(45,57)(49,59)(51,61)(56,63)(60,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,d^c=dhi,e^c=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251663);