Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418818 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15266 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,30,7,13,6,12)(3,14,29,9,18,42,17,33)(4,19,24,36,23,10,22,37)(5,11,26,39,28,41,27,40)(15,34,20,51,47,43,46,31)(16,38,25,57,49,44,48,32)(21,35,54,61,55,50,56,62)(45,59,52,60,53,63,64,58);(1,3)(2,9)(4,20)(5,25)(6,29)(7,18)(8,17)(10,34)(11,38)(12,42)(13,33)(14,30)(15,24)(16,26)(19,43)(21,53)(22,47)(23,46)(27,49)(28,48)(31,37)(32,39)(35,60)(36,51)(40,57)(41,44)(45,55)(50,58)(52,54)(56,64)(59,61)(62,63);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,31)(11,35)(12,36)(13,19)(14,43)(16,45)(17,46)(18,47)(20,29)(25,52)(26,54)(27,56)(28,55)(30,37)(32,58)(33,51)(34,42)(38,59)(39,61)(40,62)(41,50)(44,63)(48,64)(49,53)(57,60);(1,5)(2,11)(3,16)(4,21)(6,27)(7,28)(8,26)(9,32)(10,35)(12,40)(13,41)(14,44)(15,45)(17,48)(18,49)(19,50)(20,52)(22,56)(23,55)(24,54)(25,29)(30,39)(31,58)(33,57)(34,59)(36,62)(37,61)(38,42)(43,63)(46,64)(47,53)(51,60)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326014);