Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499269 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23573 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13318 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,18,34)(4,19,22,53)(5,25,27,42)(6,13,7,12)(10,35,38,23)(11,28,41,29)(15,44,46,63)(16,48,50,64)(17,43,30,33)(20,55,45,52)(21,40,24,54)(26,37,56,39)(31,57,60,47)(32,58,61,51)(36,49,59,62);(1,3)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,36)(11,32)(12,43)(13,33)(14,34)(15,21)(19,52)(22,45)(23,49)(24,46)(25,48)(26,57)(27,50)(28,58)(29,51)(31,37)(35,62)(38,59)(39,60)(40,63)(41,61)(42,64)(44,54)(47,56)(53,55);(1,4)(2,10)(3,15)(5,26)(6,24)(7,21)(8,22)(9,31)(11,40)(12,39)(13,37)(14,38)(16,49)(17,45)(18,46)(19,25)(20,30)(23,29)(27,56)(28,35)(32,52)(33,59)(34,60)(36,43)(41,54)(42,53)(44,48)(47,51)(50,62)(55,61)(57,58)(63,64);(4,21,22,24)(5,27)(10,37,38,39)(11,41)(15,45,46,20)(16,50)(19,40,53,54)(23,26,35,56)(25,42)(28,29)(31,59,60,36)(32,61)(44,52,63,55)(47,49,57,62)(48,64)(51,58)Polycyclic presentation:
2^9|a^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249766, #72249772, … |
| maximal normal subgroup of | #72249754, #72249766, #72249772, … |
| maximal quotient of | #72249891, #72250032, #72250131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406466);