Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495197 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1166 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)(3,13,18,36)(4,19)(5,23,14,10)(6,26)(7,12)(8,16)(9,22)(11,27)(15,39,40,62)(17,44,45,35)(20,47)(21,50,37,48)(24,30)(25,33,38,54)(28,57)(29,55)(31,49)(32,52,53,61)(34,59,60,42)(41,46)(43,56)(51,63)(58,64);(1,3,14,16,7,18,5,24)(2,8,23,36,12,30,10,13)(4,15,37,41,20,40,21,51)(6,17,38,43,27,45,25,55)(9,28,53,60,31,58,32,34)(11,29,54,44,26,56,33,35)(19,46,50,62,47,63,48,39)(22,42,61,57,49,59,52,64);(1,4,6,22,7,20,27,49)(2,9,11,19,12,31,26,47)(3,15,17,42,18,40,45,59)(5,21,25,52,14,37,38,61)(8,28,29,46,30,58,56,63)(10,32,33,48,23,53,54,50)(13,34,35,39,36,60,44,62)(16,41,43,57,24,51,55,64);(4,20)(9,31)(15,40)(19,47)(21,37)(22,49)(28,58)(32,53)(34,60)(39,62)(41,51)(42,59)(46,63)(48,50)(52,61)(57,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303765, #72303775, #72303792, … |
| maximal normal subgroup of | #72303765, #72303775, #72303792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402393);