Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449490 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15947 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3168 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)(3,14)(4,19,23,40)(5,11)(6,28)(7,12)(8,13)(9,17)(10,22,36,21)(15,34,46,30)(16,43)(18,32)(20,52,56,51)(24,57)(25,41)(26,38)(27,39)(29,33)(31,49)(35,55,62,54)(37,47)(42,45,50,48)(44,59,64,60)(53,61,58,63);(1,3)(2,9)(4,15)(5,24)(6,29)(7,17)(8,18)(10,34)(11,37)(12,32)(13,33)(14,28)(16,25)(19,50)(20,53)(21,48)(22,45)(23,46)(26,47)(27,49)(30,36)(31,38)(35,61)(39,43)(40,42)(41,57)(44,54)(51,59)(52,60)(55,64)(56,58)(62,63);(1,4)(2,10)(3,15)(5,20)(6,22)(7,21)(8,23)(9,30)(11,35)(12,40)(13,36)(14,42)(16,44)(17,48)(18,46)(19,28)(24,58)(25,55)(26,54)(27,56)(29,45)(31,59)(32,50)(33,34)(37,61)(38,51)(39,62)(41,52)(43,63)(47,64)(49,53)(57,60);(1,5)(2,11)(3,16)(4,20)(6,26)(7,25)(8,27)(9,31)(10,35)(12,41)(13,39)(14,43)(15,44)(17,49)(18,47)(19,51)(21,55)(22,54)(23,56)(24,29)(28,38)(30,59)(32,37)(33,57)(34,60)(36,62)(40,52)(42,63)(45,58)(46,64)(48,53)(50,61)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72263001, #72317848, #72318546, … |
| maximal normal subgroup of | #72263001, #72317848, #72318546, … |
| maximal quotient of | #72267278 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356686);