Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460137 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1149 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)(3,9,17,31)(4,10)(5,23)(6,14)(7,13)(8,12)(11,24)(15,42,44,29)(16,46,48,59)(18,39,26,32)(19,33,45,57)(20,52)(21,41)(22,35)(25,40)(27,38)(28,36)(30,49,58,50)(34,54)(37,53)(43,60,47,62)(51,56,64,63)(55,61);(1,3)(2,9,12,31)(4,19)(5,16)(6,26)(7,18)(8,17)(10,33,35,57)(11,30,38,58)(13,39,14,32)(15,21)(20,51)(22,45)(23,46,40,59)(24,50)(25,48)(27,49)(28,44)(29,41,42,36)(34,60,37,62)(43,54)(47,53)(52,56,61,63)(55,64);(1,4)(2,10)(3,15,17,44)(5,20)(6,21)(7,28)(8,22)(9,29,31,42)(11,37)(12,35)(13,36)(14,41)(16,47,48,43)(18,19,26,45)(23,52)(24,53)(25,55)(27,54)(30,56,58,63)(32,33,39,57)(34,38)(40,61)(46,60,59,62)(49,51,50,64);(1,5,8,25)(2,11)(3,16,17,48)(4,20,22,55)(6,27,7,24)(9,30)(10,34)(12,38)(13,40)(14,23)(15,43,44,47)(18,50,26,49)(19,51,45,64)(21,54,28,53)(29,56)(31,58)(32,46)(33,60)(35,37)(36,52)(39,59)(41,61)(42,63)(57,62)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=efi,g^2=i,h^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,g^b=gi,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, 367334);