Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456654 |
| 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 | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1373 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)(3,8,15,29)(4,9)(5,20)(6,23)(7,11)(10,22)(12,26)(13,31,40,27)(14,42,44,58)(16,47,46,30)(17,50)(18,37)(19,33)(21,38)(24,34)(25,36)(28,49,56,45)(32,52)(35,53)(39,61,43,64)(41,55,48,57)(51,60)(54,62,59,63);(1,3)(2,8,11,29)(4,13)(5,14)(6,16)(7,15)(9,31,33,27)(10,28,36,56)(12,30,23,47)(17,39)(18,41)(19,40)(20,42,38,58)(21,44)(22,45)(24,48)(25,49)(26,46)(32,59,35,54)(34,57,37,55)(43,51)(50,61,60,64)(52,62)(53,63);(1,4,7,19)(2,9,11,33)(3,13,15,40)(5,17,21,51)(6,24,26,18)(8,27,29,31)(10,35,36,32)(12,37,23,34)(14,43,44,39)(16,48,46,41)(20,50,38,60)(22,53,25,52)(28,54,56,59)(30,57,47,55)(42,61,58,64)(45,62,49,63);(1,5)(2,10)(3,14)(4,17)(6,25)(7,21)(8,28)(9,32)(11,36)(12,38)(13,39)(15,44)(16,49)(18,53)(19,51)(20,23)(22,26)(24,52)(27,54)(29,56)(30,58)(31,59)(33,35)(34,60)(37,50)(40,43)(41,63)(42,47)(45,46)(48,62)(55,61)(57,64)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^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,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285868, #72286090, #72286106, … |
| maximal normal subgroup of | #72285868, #72286090, #72286106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363851);