Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16584 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,19,34,31,35)(4,20,24,10,25,38,23,39)(5,26,29,42,30,41,28,11)(16,32,21,36,46,59,45,58)(17,47,50,43,51,33,49,55)(22,52,27,40,53,37,54,62)(44,61,48,64,56,60,63,57);(1,3,7,31,8,19,6,18)(2,9,13,34,14,35,12,15)(4,21,23,16,25,45,24,46)(5,17,28,49,30,51,29,50)(10,36,38,59,39,58,20,32)(11,33,26,55,42,47,41,43)(22,48,54,44,53,63,27,56)(37,61,62,64,52,60,40,57);(1,4,6,23,8,25,7,24)(2,10,12,38,14,39,13,20)(3,16,18,45,19,46,31,21)(5,27,28,53,30,54,29,22)(9,32,15,36,35,59,34,58)(11,40,26,52,42,62,41,37)(17,48,49,56,51,63,50,44)(33,60,55,64,47,61,43,57);(1,5)(2,11,14,42)(3,17)(4,22)(6,29)(7,28)(8,30)(9,33,35,47)(10,37,39,52)(12,41,13,26)(15,43,34,55)(16,44)(18,50)(19,51)(20,40,38,62)(21,48)(23,54)(24,27)(25,53)(31,49)(32,57,59,64)(36,61,58,60)(45,63)(46,56)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=e,d^2=efi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374891);