Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14517 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,17,9,18,34,30,33)(4,10,22,37,23,39,21,38)(5,24,26,11,28,41,27,42)(15,35,46,58,47,43,19,31)(16,48,50,44,49,32,25,40)(20,52,54,36,55,61,56,62)(45,57,64,59,53,60,51,63);(1,3)(2,9)(4,19)(5,25)(6,30)(7,17)(8,18)(10,35)(11,40)(12,34)(13,33)(14,29)(15,22)(16,26)(20,53)(21,47)(23,46)(24,32)(27,49)(28,50)(31,38)(36,60)(37,58)(39,43)(41,48)(42,44)(45,55)(51,54)(52,59)(56,64)(57,62)(61,63);(1,4,7,22,8,23,6,21)(2,10,12,37,13,39,29,38)(3,15,17,46,18,47,30,19)(5,20,26,54,28,55,27,56)(9,31,34,35,33,58,14,43)(11,36,41,61,42,62,24,52)(16,45,50,64,49,53,25,51)(32,57,40,59,48,60,44,63);(1,5)(2,11)(3,16)(4,20)(6,27)(7,26)(8,28)(9,32)(10,36)(12,41)(13,42)(14,44)(15,45)(17,50)(18,49)(19,51)(21,56)(22,54)(23,55)(24,29)(25,30)(31,57)(33,48)(34,40)(35,59)(37,61)(38,52)(39,62)(43,63)(46,64)(47,53)(58,60)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295382);