Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,26,7,12,30,11)(3,13,27,33,17,38,16,8)(4,18,20,35,21,9,28,39)(5,22,24,37,25,10,29,40)(14,41,47,31,45,57,44,49)(15,36,48,58,46,42,23,32)(19,34,52,61,53,50,54,60)(43,62,55,56,63,59,51,64);(1,3)(2,8)(4,14)(5,23)(6,27)(7,17)(9,31)(10,36)(11,38)(12,33)(13,26)(15,24)(16,30)(18,49)(19,51)(20,47)(21,45)(22,42)(25,48)(28,44)(29,46)(32,37)(34,59)(35,41)(39,57)(40,58)(43,52)(50,62)(53,55)(54,63)(56,60)(61,64);(1,4,7,21)(2,9,12,18)(3,14,17,45)(5,19,25,53)(6,28,30,20)(8,31,33,49)(10,34,22,50)(11,39,26,35)(13,41,38,57)(15,43,46,63)(16,47,27,44)(23,55,48,51)(24,54,29,52)(32,56,58,64)(36,62,42,59)(37,61,40,60);(1,5)(2,10)(3,15)(4,19)(6,29)(7,25)(8,32)(9,34)(11,40)(12,22)(13,42)(14,43)(16,48)(17,46)(18,50)(20,54)(21,53)(23,27)(24,30)(26,37)(28,52)(31,56)(33,58)(35,61)(36,38)(39,60)(41,59)(44,55)(45,63)(47,51)(49,64)(57,62)Polycyclic presentation:
2^9|a^2=ef,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326069);