Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419800 |
| 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)(9,10,14,20,16,21,15,38)(11,22,25,41,27,49,26,17)(12,18,30,44,32,45,31,62)(13,33,36,19,37,47,35,48)(23,42,28,56,53,67,52,39)(24,40,54,50,55,65,34,46)(29,58,60,43,61,68,59,69)(51,64,57,66,71,72,63,70);(9,11,16,27)(10,17,21,41)(12,28,32,52)(13,34,37,54)(14,26,15,25)(18,42,45,67)(19,46,48,50)(20,49,38,22)(23,31,53,30)(24,36,55,35)(29,51,61,71)(33,40,47,65)(39,62,56,44)(43,64,69,72)(57,60,63,59)(58,66,68,70);(2,5)(6,8)(9,12,15,31,16,32,14,30)(10,18,38,62,21,45,20,44)(11,23,26,52,27,53,25,28)(13,29,35,59,37,61,36,60)(17,39,49,67,41,56,22,42)(19,43,33,58,48,69,47,68)(24,51,34,63,55,71,54,57)(40,64,46,70,65,72,50,66);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,29)(14,36)(15,35)(16,37)(17,40)(18,43)(20,47)(21,48)(22,50)(23,51)(25,54)(26,34)(27,55)(28,57)(30,60)(31,59)(32,61)(33,38)(39,64)(41,65)(42,66)(44,68)(45,69)(46,49)(52,63)(53,71)(56,72)(58,62)(67,70)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=f,e^2=i,g^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^b=gi,h^b=hi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 352319);