Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386374 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14427 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,19,33,20,9,18,34)(4,21,25,37,24,41,30,10)(5,26,27,11,29,40,28,39)(16,43,22,31,45,35,50,42)(17,46,47,38,49,57,48,32)(23,36,56,51,54,61,55,60)(44,62,52,63,64,58,53,59);(1,3,8,20)(2,9,12,15)(4,22,24,50)(5,17,29,49)(6,19,7,18)(10,35,37,43)(11,38,39,32)(13,34,14,33)(16,30,45,25)(21,31,41,42)(23,53,54,52)(26,57,40,46)(27,48,28,47)(36,59,61,63)(44,56,64,55)(51,58,60,62);(1,4)(2,10,12,37)(3,16)(5,23)(6,25)(7,30)(8,24)(9,31,15,42)(11,36,39,61)(13,21,14,41)(17,44)(18,22)(19,50)(20,45)(26,51,40,60)(27,55)(28,56)(29,54)(32,58,38,62)(33,35,34,43)(46,63,57,59)(47,53)(48,52)(49,64);(1,5)(2,11)(3,17)(4,23)(6,28)(7,27)(8,29)(9,32)(10,36)(12,39)(13,26)(14,40)(15,38)(16,44)(18,48)(19,47)(20,49)(21,51)(22,52)(24,54)(25,56)(30,55)(31,58)(33,57)(34,46)(35,59)(37,61)(41,60)(42,62)(43,63)(45,64)(50,53)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293573);