Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10833 |
| prime radical | C24 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,14,28,43,8,13,29,42,5,11,26,40)(3,15,19,45,20,9,30,33,18,34,27,56,17,32,49,41)(4,10,23,37,22,36,52,60,24,38,54,62,25,39,55,63)(16,35,47,59,21,50,51,64,46,44,57,61,48,31,53,58);(1,3)(2,9,13,32)(4,21,24,48)(5,17)(6,27)(7,20)(8,18)(10,35)(11,15,14,34)(12,41,42,33)(16,22,46,25)(19,29)(23,53,54,51)(26,30)(28,49)(31,39)(36,50)(37,61)(38,44)(40,56,43,45)(47,55,57,52)(58,60)(59,62)(63,64);(1,4)(2,10)(3,16,18,46)(5,25)(6,23)(7,22)(8,24)(9,31,32,50)(11,39)(12,37)(13,38)(14,36)(15,44,34,35)(17,48,20,21)(19,47,27,57)(26,55)(28,52)(29,54)(30,51,49,53)(33,58,41,64)(40,63)(42,62)(43,60)(45,61,56,59)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,e^b=ei,f^b=fi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265417, #72265425, #72265441, … |
| maximal normal subgroup of | #72265417, #72265425, #72265441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54154);