Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3884 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)(3,15)(4,21)(5,25)(6,14)(7,12)(8,13)(9,18)(10,23)(11,26)(16,34)(17,31)(19,33)(20,32)(22,35)(24,38)(27,39)(28,36)(29,37)(30,45)(40,44)(41,47)(42,60)(43,49)(46,54)(48,57)(50,52)(51,55)(53,59)(56,62)(58,61)(63,64);(1,3)(2,9)(4,16)(5,17)(6,19)(7,18)(8,20)(10,34)(11,31)(12,32)(13,33)(14,15)(21,49)(22,42)(23,44)(24,45)(25,54)(26,47)(27,48)(28,43)(29,46)(30,36)(35,58)(37,57)(38,40)(39,41)(50,64)(51,62)(52,61)(53,63)(55,60)(56,59);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,30)(11,35)(12,38)(13,36)(14,21)(15,40)(17,42)(18,44)(19,43)(20,45)(25,55)(26,52)(27,53)(29,51)(31,56)(32,49)(33,34)(37,59)(39,50)(41,60)(46,62)(47,61)(48,63)(54,64)(57,58);(1,5,8,27)(2,11,13,37)(3,17,20,48)(4,22,24,53)(6,26,7,29)(9,31,33,57)(10,35,36,59)(12,25,14,39)(15,41,32,54)(16,42,45,63)(18,46,19,47)(21,50,38,55)(23,51,28,52)(30,56,34,58)(40,60,49,64)(43,61,44,62)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251694, #72251718, #72251726, … |
| maximal normal subgroup of | #72251694, #72251718, #72251726, … |
| maximal quotient of | #72255910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301303);