Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16284 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,13,36)(3,8,17,29,16,30,25,37)(4,18,21,32,22,38,26,9)(5,10,23,35,24,34,39,59)(14,40,42,55,19,47,46,27)(15,28,45,56,44,57,52,60)(20,49,50,31,51,48,53,33)(41,62,63,54,43,61,64,58);(1,3,13,17,7,16,6,25)(2,8,12,37,11,30,36,29)(4,19,26,46,22,14,21,42)(5,15,39,45,24,44,23,52)(9,27,18,47,32,55,38,40)(10,28,35,60,34,57,59,56)(20,43,53,64,51,41,50,63)(31,54,48,62,33,58,49,61);(1,4)(2,9)(3,14)(5,20)(6,26)(7,22)(8,27)(10,33)(11,32)(12,38)(13,21)(15,43)(16,19)(17,46)(18,36)(23,53)(24,51)(25,42)(28,54)(29,47)(30,55)(31,34)(35,48)(37,40)(39,50)(41,44)(45,63)(49,59)(52,64)(56,62)(57,58)(60,61);(1,5)(2,10)(3,15)(4,20)(6,23)(7,24)(8,28)(9,31)(11,34)(12,35)(13,39)(14,41)(16,44)(17,45)(18,48)(19,43)(21,50)(22,51)(25,52)(26,53)(27,54)(29,56)(30,57)(32,33)(36,59)(37,60)(38,49)(40,61)(42,63)(46,64)(47,62)(55,58)Polycyclic presentation:
2^9|a^2=f,b^2=efi,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,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, 363865);