Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436179 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15550 |
| 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 | |
912 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)(3,13,17,37)(4,18,23,9)(5,24)(6,11)(7,12)(8,16,31,28)(10,27)(14,38)(15,41,42,35)(19,46)(20,32)(21,33,53,44)(22,45,49,34)(25,30,56,58)(26,54)(29,48)(36,50)(39,61)(40,43)(47,59)(51,63)(52,60,62,64)(55,57);(1,3)(2,8)(4,19)(5,25)(6,28)(7,17)(9,29)(10,35)(11,37)(12,31)(13,32)(14,23)(15,26)(16,20)(18,43)(21,51)(22,40)(24,41)(27,56)(30,36)(33,59)(34,46)(38,45)(39,62)(42,50)(44,61)(47,52)(48,49)(53,55)(54,58)(57,60)(63,64);(1,4,20,49,7,23,6,22)(2,9,32,45,12,18,11,34)(3,14,28,48,17,19,16,40)(5,21,50,62,27,53,26,52)(8,29,37,46,31,43,13,38)(10,33,54,64,24,44,36,60)(15,39,56,51,42,47,25,55)(30,57,41,61,58,63,35,59);(1,5)(2,10,12,24)(3,15,17,42)(4,21)(6,26)(7,27)(8,30)(9,33,18,44)(11,36,32,54)(13,35)(14,39,19,47)(16,25,28,56)(20,50)(22,52)(23,53)(29,57)(31,58)(34,60,45,64)(37,41)(38,59)(40,55,48,51)(43,63)(46,61)(49,62)Polycyclic presentation:
2^9|a^2=fg,c^2=ei,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343376);