Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1519 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8023 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,13,7,29)(3,14,17,38,19,41,18,42)(4,10,21,36,22,37,23,53)(5,24,26,39,27,54,28,11)(9,30,32,47,34,25,33,16)(15,40,44,59,45,63,46,62)(20,48,50,60,52,61,51,35)(31,55,56,64,57,49,58,43);(1,3)(2,9)(4,15,22,45)(5,25,27,30)(6,17)(7,18)(8,19)(10,31,37,57)(11,38,39,42)(12,32)(13,34)(14,24,41,54)(16,28,47,26)(20,49)(21,44,23,46)(29,33)(35,59)(36,56,53,58)(40,61)(43,50)(48,63)(51,64)(52,55)(60,62);(1,4)(2,10)(3,15)(5,20)(6,21)(7,23)(8,22)(9,31)(11,35)(12,36)(13,37)(14,40)(16,43)(17,44)(18,46)(19,45)(24,48)(25,49)(26,50)(27,52)(28,51)(29,53)(30,55)(32,56)(33,58)(34,57)(38,59)(39,60)(41,63)(42,62)(47,64)(54,61)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280848, #72280909, #72280958, … |
| maximal normal subgroup of | #72280848, #72280909, #72280958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34827);