Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16697 |
| 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 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,27)(3,14,18,33,17,9,29,42)(4,10,30,36,21,37,22,52)(5,23,28,39,25,11,26,38)(15,31,19,48,44,40,50,34)(16,32,46,59,45,41,47,58)(20,35,53,61,24,54,51,62)(43,56,55,60,63,57,49,64);(1,3)(2,9,13,14)(4,19,21,50)(5,16)(6,17)(7,18)(8,29)(10,34)(11,32,23,41)(12,33,27,42)(15,22,44,30)(20,49,24,55)(25,45)(26,46)(28,47)(31,52)(35,60)(36,40)(37,48)(38,58,39,59)(43,53,63,51)(54,64)(56,61)(57,62);(1,4)(2,10)(3,15,17,44)(5,24)(6,21)(7,22)(8,30)(9,31,14,40)(11,35)(12,36)(13,37)(16,43,45,63)(18,19,29,50)(20,25)(23,54)(26,53)(27,52)(28,51)(32,57,41,56)(33,34,42,48)(38,61)(39,62)(46,55,47,49)(58,60,59,64);(1,5,6,25)(2,11,13,23)(3,16,17,45)(4,20,21,24)(7,28,8,26)(9,32,14,41)(10,35,37,54)(12,39,27,38)(15,43,44,63)(18,47,29,46)(19,49,50,55)(22,53,30,51)(31,56,40,57)(33,59,42,58)(34,60,48,64)(36,62,52,61)Polycyclic presentation:
2^9|a^2=eg,b^2=h,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377939);