Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15257 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2328 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)(3,9)(4,19)(5,24)(6,13)(7,12)(8,14)(10,23)(11,25)(15,41)(16,45)(17,33)(18,34)(20,30)(21,36)(22,38)(26,39)(27,40)(28,32)(29,37)(31,47)(35,43)(42,63)(44,49)(46,59)(48,58)(50,55)(51,61)(52,60)(53,62)(54,56)(57,64);(1,3)(2,9)(4,20)(5,16)(6,28)(7,17)(8,18)(10,35)(11,31)(12,33)(13,32)(14,34)(15,23)(19,49)(21,52)(22,44)(24,45)(25,47)(26,46)(27,48)(29,43)(30,38)(36,61)(37,41)(39,59)(40,58)(42,55)(50,57)(51,53)(54,64)(56,63)(60,62);(1,4)(2,10)(3,15)(5,21)(6,29)(7,23)(8,22)(9,30)(11,36)(12,38)(13,19)(14,37)(16,42)(17,44)(18,43)(20,28)(24,56)(25,55)(26,54)(27,53)(31,57)(32,35)(33,41)(34,49)(39,62)(40,50)(45,61)(46,52)(47,51)(48,64)(58,60)(59,63);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,27)(9,31)(10,36)(12,24)(13,40)(14,39)(15,42)(17,46)(18,48)(19,50)(20,51)(22,53)(23,54)(28,47)(29,55)(30,57)(32,58)(33,45)(34,59)(35,60)(37,62)(38,56)(41,61)(43,64)(44,52)(49,63)Polycyclic presentation:
2^9|f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324826);