Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15561 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,8,12)(3,15,19,33,20,41,18,9)(4,10,23,37,24,38,30,36)(5,25,29,40,28,42,27,11)(16,34,21,50,46,43,45,31)(17,47,26,39,49,32,48,57)(22,52,55,35,54,60,53,61)(44,56,63,62,64,59,51,58);(1,3,8,18,7,20,6,19)(2,9,12,41,14,33,13,15)(4,21,30,16,24,45,23,46)(5,26,29,49,28,48,27,17)(10,34,36,31,38,43,37,50)(11,39,25,32,40,57,42,47)(22,44,55,63,54,64,53,51)(35,59,60,58,61,56,52,62);(1,4)(2,10)(3,16)(5,22)(6,30)(7,24)(8,23)(9,31)(11,35)(12,37)(13,36)(14,38)(15,43)(17,44)(18,21)(19,45)(20,46)(25,52)(26,51)(27,55)(28,54)(29,53)(32,56)(33,50)(34,41)(39,62)(40,61)(42,60)(47,59)(48,63)(49,64)(57,58);(1,5,7,28)(2,11)(3,17)(4,22,24,54)(6,27,8,29)(9,32,33,47)(10,35)(12,25)(13,42)(14,40)(15,39,41,57)(16,44)(18,26)(19,48)(20,49)(21,51)(23,53,30,55)(31,56,50,59)(34,58,43,62)(36,60)(37,52)(38,61)(45,63)(46,64)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343667);