Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21452 |
| prime radical | #33530 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,15,31,14)(3,16,21,42,20,45,30,40)(4,10,24,37,25,39,58,38)(5,11,6,27,26,12,29,41)(9,18,34,19,33,51,35,53)(17,46,50,63,49,44,57,64)(22,36,28,59,54,43,23,55)(32,56,61,48,60,47,62,52);(1,3)(2,9)(4,17)(5,19)(6,18)(7,30)(8,20)(10,32)(11,40)(12,42)(13,35)(14,34)(15,33)(16,41)(21,31)(22,52)(23,56)(24,57)(25,49)(26,53)(27,45)(28,47)(29,51)(36,63)(37,62)(38,61)(39,60)(43,64)(44,55)(46,59)(48,54)(50,58);(1,4)(2,10)(3,17)(5,22)(6,28)(7,24)(8,25)(9,32)(11,36)(12,43)(13,37)(14,38)(15,39)(16,44)(18,47)(19,52)(20,49)(21,50)(23,29)(26,54)(27,59)(30,57)(31,58)(33,60)(34,61)(35,62)(40,63)(41,55)(42,64)(45,46)(48,53)(51,56);(1,5,7,6,8,26,31,29)(2,11,13,27,15,12,14,41)(3,18,21,19,20,51,30,53)(4,22,24,28,25,54,58,23)(9,16,34,42,33,45,35,40)(10,36,37,59,39,43,38,55)(17,47,50,52,49,56,57,48)(32,44,61,64,60,46,62,63)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280077, #72281057, #72281216, … |
| maximal normal subgroup of | #72280077, #72281057, #72281216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402931);