Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32947 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14698 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,19,32,18,39,17,9)(4,20,23,34,24,49,22,10)(5,25,27,36,28,11,26,35)(15,40,44,59,43,30,42,37)(16,38,46,31,47,57,45,54)(21,50,52,55,53,48,51,33)(41,60,62,58,63,56,61,64);(1,3)(2,9,13,32)(4,15)(5,16)(6,17)(7,19)(8,18)(10,30,34,40)(11,31,25,54)(12,14,29,39)(20,37,49,59)(21,41)(22,42)(23,44)(24,43)(26,45)(27,46)(28,47)(33,56,55,60)(35,57,36,38)(48,58,50,64)(51,61)(52,62)(53,63);(1,4,6,22,8,24,7,23)(2,10,12,20,13,34,29,49)(3,15,17,42,18,43,19,44)(5,21,27,52,28,53,26,51)(9,30,14,37,32,40,39,59)(11,33,36,48,25,55,35,50)(16,41,46,62,47,63,45,61)(31,56,38,58,54,60,57,64);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,31)(10,33)(12,36)(13,25)(14,38)(15,41)(17,46)(18,47)(19,45)(20,48)(22,52)(23,51)(24,53)(29,35)(30,56)(32,54)(34,55)(37,58)(39,57)(40,60)(42,62)(43,63)(44,61)(49,50)(59,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,g^b=gi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281928, #72282021, #72282115, … |
| maximal normal subgroup of | #72281928, #72282021, #72282115, … |
| maximal quotient of | #72343107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300228);