Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16808 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,12,7,14)(3,9,18,32,17,33,19,41)(4,20,23,36,25,37,24,10)(5,26,27,39,29,40,28,11)(15,34,44,56,43,59,21,30)(16,45,48,31,47,57,49,58)(22,38,52,61,54,35,53,50)(42,55,63,60,46,64,51,62);(1,3)(2,9,12,33)(4,21)(5,16)(6,19)(7,18)(8,17)(10,34,36,59)(11,31,39,58)(13,41,14,32)(15,24)(20,30,37,56)(22,51)(23,43)(25,44)(26,45,40,57)(27,49)(28,48)(29,47)(35,60,38,62)(42,53)(46,52)(50,55,61,64)(54,63);(1,4,8,25)(2,10,12,36)(3,15,17,43)(5,22,29,54)(6,24,7,23)(9,30,33,56)(11,38,39,35)(13,37,14,20)(16,46,47,42)(18,21,19,44)(26,50,40,61)(27,53,28,52)(31,55,58,64)(32,59,41,34)(45,60,57,62)(48,63,49,51);(1,5)(2,11)(3,16)(4,22)(6,27)(7,28)(8,29)(9,31)(10,35)(12,39)(13,26)(14,40)(15,42)(17,47)(18,48)(19,49)(20,50)(21,51)(23,52)(24,53)(25,54)(30,55)(32,57)(33,58)(34,60)(36,38)(37,61)(41,45)(43,46)(44,63)(56,64)(59,62)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72286957, #72287280, #72287312, … |
| maximal normal subgroup of | #72286957, #72287280, #72287312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379093);