Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393982 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(9,10,16,22)(11,23,28,39)(12,29,33,42)(13,19,35,45)(14,21,15,20)(17,27,40,26)(18,31,43,32)(24,49,53,37)(25,54,56,48)(30,59,61,66)(34,46,36,44)(38,57,64,55)(41,62,65,60)(47,51,58,52)(50,63,70,68)(67,69,72,71);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,26)(16,28)(18,37)(19,38)(20,39)(21,23)(22,40)(29,58)(30,50)(31,52)(32,51)(33,53)(34,57)(35,56)(36,55)(41,63)(42,47)(43,49)(44,48)(45,64)(46,54)(59,72)(60,71)(61,70)(62,69)(65,68)(66,67);(2,5)(6,8)(9,12,16,33)(10,18,22,43)(11,24,28,53)(13,30,35,61)(14,32,15,31)(17,37,40,49)(19,41,45,65)(20,29,21,42)(23,47,39,58)(25,50,56,70)(26,52,27,51)(34,62,36,60)(38,63,64,68)(44,66,46,59)(48,67,54,72)(55,71,57,69);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,25)(12,30)(14,36)(15,34)(16,35)(17,38)(18,41)(20,46)(21,44)(22,45)(23,48)(24,50)(26,57)(27,55)(28,56)(29,59)(31,62)(32,60)(33,61)(37,63)(39,54)(40,64)(42,66)(43,65)(47,67)(49,68)(51,71)(52,69)(53,70)(58,72)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311192);