Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460120 |
| 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 | |
1148 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)(3,8,15,27)(4,9)(5,20,21,51)(6,12)(7,11)(10,22,33,52)(13,36,38,25)(14,40)(16,28,23,35)(17,44,45,50)(18,30)(19,31)(24,34)(26,53)(29,46,32,59)(37,61)(39,58,48,55)(41,63)(42,56)(43,49)(47,57)(54,62)(60,64);(1,3)(2,8,11,27)(4,13,18,38)(5,14)(6,23)(7,15)(9,25)(10,26,33,56)(12,35,34,28)(16,24)(17,37,45,41)(19,48,47,39)(20,49,51,40)(21,43)(22,53)(29,54)(30,36)(31,58)(32,60)(42,52)(44,61)(46,64,59,62)(50,63)(55,57);(1,4)(2,9)(3,13,15,38)(5,17,21,45)(6,19)(7,18)(8,25,27,36)(10,32,33,29)(11,30)(12,31)(14,41)(16,39,23,48)(20,50,51,44)(22,46,52,59)(24,47)(26,54)(28,55,35,58)(34,57)(37,43)(40,63)(42,64)(49,61)(53,62)(56,60);(1,5,7,21)(2,10)(3,14,15,43)(4,17)(6,22,24,52)(8,26)(9,29,30,32)(11,33)(12,20)(13,37)(16,42,23,53)(18,45)(19,46)(25,54,36,60)(27,56)(28,40)(31,44,57,50)(34,51)(35,49)(38,41)(39,62)(47,59)(48,64)(55,63,58,61)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367317);