Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,22,33)(3,8,15,26,16,25,39,50)(4,9,19,30,18,29,43,45)(5,10,21,32,20,31,44,56)(13,23,36,47,35,27,51,53)(14,24,38,49,37,48,59,63)(17,40,42,60,41,28,54,55)(34,52,58,64,57,46,61,62);(1,3,6,15,7,16,22,39)(2,8,12,26,11,25,33,50)(4,13,19,36,18,35,43,51)(5,14,21,38,20,37,44,59)(9,27,30,53,29,23,45,47)(10,24,32,49,31,48,56,63)(17,34,42,58,41,57,54,61)(28,52,55,64,40,46,60,62);(1,4)(2,9)(3,13)(5,17)(6,19)(7,18)(8,23)(10,28)(11,29)(12,30)(14,34)(15,36)(16,35)(20,41)(21,42)(22,43)(24,46)(25,27)(26,47)(31,40)(32,55)(33,45)(37,57)(38,58)(39,51)(44,54)(48,52)(49,62)(50,53)(56,60)(59,61)(63,64);(1,5)(2,10)(3,14)(4,17)(6,21)(7,20)(8,24)(9,28)(11,31)(12,32)(13,34)(15,38)(16,37)(18,41)(19,42)(22,44)(23,46)(25,48)(26,49)(27,52)(29,40)(30,55)(33,56)(35,57)(36,58)(39,59)(43,54)(45,60)(47,62)(50,63)(51,61)(53,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317346, #72317348, #72317353, … |
| maximal normal subgroup of | #72317346, #72317348, #72317353, … |
| maximal quotient of | C2×#364501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271697);