Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
868 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)(3,6)(4,5)(7,8)(9,10,14,21,16,22,15,20)(11,23,26,17,28,42,27,43)(12,29,32,47,34,46,33,18)(13,35,39,50,38,49,37,19)(24,53,30,44,56,40,55,51)(25,41,58,48,57,65,36,52)(31,61,64,66,63,45,62,59)(54,68,60,67,72,69,71,70);(2,5)(3,7)(9,11,14,26,16,28,15,27)(10,17,20,23,22,43,21,42)(12,30,32,56,34,55,33,24)(13,36,37,57,38,58,39,25)(18,44,46,53,47,51,29,40)(19,48,35,65,50,52,49,41)(31,54,62,71,63,72,64,60)(45,67,59,69,61,70,66,68);(9,12,16,34)(10,18,22,47)(11,24,28,56)(13,31,38,63)(14,33,15,32)(17,40,43,53)(19,45,50,61)(20,29,21,46)(23,51,42,44)(25,54,57,72)(26,55,27,30)(35,66,49,59)(36,60,58,71)(37,64,39,62)(41,67,65,70)(48,68,52,69);(1,3,4,7)(2,6,5,8)(9,13,16,38)(10,19)(11,25,28,57)(12,31,34,63)(14,39,15,37)(17,41)(18,45)(20,49)(21,35)(22,50)(23,52)(24,54,56,72)(26,58,27,36)(29,59)(30,60,55,71)(32,64,33,62)(40,67)(42,48)(43,65)(44,68)(46,66)(47,61)(51,69)(53,70)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=i,d^2=ef,e^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,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 360178);