Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419794 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
923 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)(9,10,14,21,16,20,15,36)(11,22,25,48,27,17,26,40)(12,18,30,44,32,43,31,62)(13,33,34,19,35,47,37,46)(23,49,28,57,52,38,51,41)(24,53,54,45,55,64,56,39)(29,59,60,42,61,69,63,68)(50,66,58,70,71,67,72,65);(9,11,16,27)(10,17,20,22)(12,28,32,51)(13,24,35,55)(14,26,15,25)(18,41,43,57)(19,45,46,39)(21,48,36,40)(23,31,52,30)(29,58,61,72)(33,64,47,53)(34,56,37,54)(38,62,49,44)(42,67,68,66)(50,63,71,60)(59,65,69,70);(2,5)(6,8)(9,12,15,31,16,32,14,30)(10,18,36,62,20,43,21,44)(11,23,26,51,27,52,25,28)(13,29,37,63,35,61,34,60)(17,38,48,57,22,49,40,41)(19,42,33,59,46,68,47,69)(24,50,56,72,55,71,54,58)(39,65,64,67,45,70,53,66);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,29)(14,34)(15,37)(16,35)(17,39)(18,42)(20,46)(21,47)(22,45)(23,50)(25,54)(26,56)(27,55)(28,58)(30,60)(31,63)(32,61)(33,36)(38,65)(40,53)(41,66)(43,68)(44,69)(48,64)(49,70)(51,72)(52,71)(57,67)(59,62)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 352325);