Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461183 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16382 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
857 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)(3,13,16,8)(4,9,20,34)(5,22)(6,25)(7,11)(10,23)(12,28)(14,32)(15,44,45,55)(17,31,47,41)(18,29)(19,51,52,60)(21,38,26,35)(24,39)(27,37)(30,46,40,49)(33,54,36,53)(42,61)(43,57)(48,59)(50,64)(56,63)(58,62);(1,3)(2,8)(4,18,20,14)(5,15)(6,17)(7,16)(9,32,34,29)(10,30)(11,13)(12,31)(19,50,52,42)(21,48,26,43)(22,55)(23,46)(24,45)(25,41)(27,49)(28,47)(33,58,36,56)(35,59,38,57)(37,40)(39,44)(51,61,60,64)(53,63,54,62);(1,4)(2,9,11,34)(3,14)(5,19)(6,26)(7,20)(8,29,13,32)(10,36,37,33)(12,38,25,35)(15,42)(16,18)(17,48)(21,28)(22,51,39,60)(23,54)(24,52)(27,53)(30,58,40,56)(31,59,41,57)(43,47)(44,61,55,64)(45,50)(46,63)(49,62);(1,5,7,24)(2,10,11,37)(3,15,16,45)(4,19,20,52)(6,27,28,23)(8,30,13,40)(9,33,34,36)(12,39,25,22)(14,42,18,50)(17,49,47,46)(21,54,26,53)(29,56,32,58)(31,44,41,55)(35,60,38,51)(43,63,48,62)(57,61,59,64)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368380);