Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,23)(12,29,33,18)(13,34)(14,22)(15,20)(16,21)(17,26)(19,37)(24,44,53,60)(25,54)(27,42)(28,43)(30,40,59,69)(31,63,64,47)(32,46,39,50)(35,61,65,45)(36,49)(38,48)(41,57)(51,58)(52,68,67,72)(55,71,62,70)(56,66);(1,3,4,7)(2,5,6,8)(9,11,16,27)(10,17,21,43)(12,30,33,59)(13,25,38,57)(14,28,15,26)(18,44,29,60)(19,41,49,54)(20,42,22,23)(24,39,53,32)(31,62,64,55)(34,66,48,51)(35,67,65,52)(36,58,37,56)(40,50,69,46)(45,71,61,70)(47,72,63,68);(3,7)(5,8)(9,12,14,32,16,33,15,39)(10,18,22,50,21,29,20,46)(11,24,28,59,27,53,26,30)(13,35,36,64,38,65,37,31)(17,40,23,44,43,69,42,60)(19,47,34,45,49,63,48,61)(25,55,58,52,57,62,56,67)(41,70,66,72,54,71,51,68);(9,13,16,38)(10,19,21,49)(11,25,27,57)(12,31,33,64)(14,37,15,36)(17,41,43,54)(18,45,29,61)(20,34,22,48)(23,51,42,66)(24,52,53,67)(26,58,28,56)(30,62,59,55)(32,65,39,35)(40,68,69,72)(44,71,60,70)(46,63,50,47)Polycyclic presentation:
2^9|a^2=eg,b^2=gh,c^2=ef,d^2=i,e^2=i,g^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,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381257);