Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380130 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14259 |
| 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 | |
1097 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)(3,15,19,39,18,9,28,32)(4,20,24,35,25,10,23,36)(5,11,26,38,27,37,29,40)(16,30,21,33,44,41,49,50)(17,42,48,58,46,31,47,59)(22,51,53,62,54,34,55,61)(43,56,52,60,45,57,63,64);(1,3)(2,9,12,15)(4,21)(5,17)(6,28)(7,19)(8,18)(10,33,20,50)(11,31,37,42)(13,39,14,32)(16,23)(22,52)(24,44)(25,49)(26,48)(27,46)(29,47)(30,36,41,35)(34,60,51,64)(38,59,40,58)(43,55)(45,53)(54,63)(56,61,57,62);(1,4,8,25)(2,10,12,20)(3,16,18,44)(5,22,27,54)(6,24,7,23)(9,30,15,41)(11,34,37,51)(13,36,14,35)(17,45,46,43)(19,49,28,21)(26,55,29,53)(31,57,42,56)(32,50,39,33)(38,62,40,61)(47,63,48,52)(58,64,59,60);(1,5)(2,11)(3,17)(4,22)(6,29)(7,26)(8,27)(9,31)(10,34)(12,37)(13,40)(14,38)(15,42)(16,43)(18,46)(19,48)(20,51)(21,52)(23,55)(24,53)(25,54)(28,47)(30,56)(32,59)(33,60)(35,62)(36,61)(39,58)(41,57)(44,45)(49,63)(50,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287329);