Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14915 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,33)(4,10,24,37)(5,25,27,38)(6,13,7,12)(9,20,32,18)(11,26,39,28)(16,41,21,49)(17,31,46,58)(22,51,53,55)(23,40,29,36)(30,48,34,44)(35,56,60,52)(42,45,54,47)(43,59,50,57)(61,63,64,62);(1,3,7,20,8,19,6,18)(2,9,13,33,14,32,12,15)(4,21,23,48,24,16,29,44)(5,17,28,47,27,46,26,45)(10,34,36,49,37,30,40,41)(11,31,25,54,39,58,38,42)(22,50,52,63,53,43,56,62)(35,59,51,64,60,57,55,61);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,36)(13,40)(14,37)(15,41)(17,43)(18,44)(19,21)(20,48)(25,55)(26,52)(27,53)(28,56)(31,57)(32,34)(33,49)(38,51)(39,60)(42,61)(45,62)(46,50)(47,63)(54,64)(58,59);(1,5,6,26,8,27,7,28)(2,11,12,38,14,39,13,25)(3,17,18,45,19,46,20,47)(4,22,23,52,24,53,29,56)(9,31,15,42,32,58,33,54)(10,35,36,51,37,60,40,55)(16,43,44,62,21,50,48,63)(30,57,41,61,34,59,49,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284248, #72285951, #72286206, … |
| maximal normal subgroup of | #72284248, #72285951, #72286206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312816);