Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1073 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(2,5)(3,7)(9,10,16,22,15,21,14,20)(11,23,28,17,27,49,26,42)(12,29,39,45,33,18,32,46)(13,34,38,19,37,50,36,48)(24,52,55,43,30,51,54,40)(25,41,57,69,56,68,35,47)(31,60,64,58,63,65,62,44)(53,70,61,72,59,67,66,71);(1,2,4,5)(3,6,7,8)(9,11)(10,17)(12,30)(13,35)(14,28)(15,27)(16,26)(18,43)(19,47)(20,49)(21,42)(22,23)(24,33)(25,36)(29,40)(31,61)(32,54)(34,41)(37,57)(38,56)(39,55)(44,71)(45,51)(46,52)(48,69)(50,68)(53,64)(58,72)(59,62)(60,67)(63,66)(65,70);(1,3,4,7)(2,6,5,8)(9,12,14,32,15,33,16,39)(10,18,20,45,21,29,22,46)(11,24,26,54,27,30,28,55)(13,31,36,62,37,63,38,64)(17,40,23,51,42,43,49,52)(19,44,34,65,48,58,50,60)(25,53,35,66,56,59,57,61)(41,67,47,72,68,70,69,71);(2,5)(6,8)(9,13,14,36,15,37,16,38)(10,19,20,34,21,48,22,50)(11,25,26,35,27,56,28,57)(12,31,32,62,33,63,39,64)(17,41,23,47,42,68,49,69)(18,44,45,65,29,58,46,60)(24,53,54,66,30,59,55,61)(40,67,51,72,43,70,52,71)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337833);