Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13847 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ 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)(3,13)(4,18,21,9)(5,10)(6,24)(7,12)(8,16)(11,26)(14,36,40,55)(15,37)(17,35)(19,43,46,31)(20,32,48,44)(22,49)(23,34)(25,29)(27,47,30,39)(28,41)(33,51)(38,58,60,64)(42,57)(45,56,63,61)(50,53)(52,62,54,59);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,30)(10,28)(11,35)(12,29)(13,24)(16,26)(18,27)(19,38)(20,47)(21,40)(22,50)(23,42)(31,54)(32,36)(33,57)(34,53)(37,49)(39,48)(41,51)(43,52)(44,55)(45,62)(46,60)(56,58)(59,63)(61,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,27)(10,31)(11,32)(12,18)(13,36)(15,38)(16,39)(17,40)(22,45)(23,46)(24,44)(25,47)(26,48)(28,52)(29,30)(33,56)(34,43)(35,55)(37,58)(41,59)(42,60)(49,61)(50,62)(51,63)(53,54)(57,64);(1,5,6,22,7,23,26,51)(2,10,24,49,12,34,11,33)(3,15,25,50,17,42,16,41)(4,19,20,45,21,46,48,63)(8,28,13,37,29,53,35,57)(9,31,44,61,18,43,32,56)(14,38,47,62,40,60,39,59)(27,52,36,58,30,54,55,64)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285493, #72285525, C2×#358715, … |
| maximal normal subgroup of | #72285493, #72285525, C2×#358715, … |
| maximal quotient of | C2×#358715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265911);