Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #472681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16831 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9,17,31)(4,10)(5,24)(6,14)(7,13)(8,12)(11,26)(15,42,44,29)(16,45,47,58)(18,33,19,32)(20,34,50,59)(21,52)(22,41)(23,36)(25,39)(27,40)(28,37)(30,48,57,49)(35,53)(38,55)(43,60,46,62)(51,56,64,63)(54,61);(1,3)(2,9,12,31)(4,20,23,50)(5,16)(6,18)(7,19)(8,17)(10,34)(11,30,39,57)(13,32,14,33)(15,22,44,28)(21,51,54,64)(24,45,40,58)(25,48)(26,49)(27,47)(29,41)(35,60)(36,59)(37,42)(38,62)(43,55,46,53)(52,56)(61,63);(1,4)(2,10)(3,15,17,44)(5,21)(6,22)(7,28)(8,23)(9,29,31,42)(11,38)(12,36)(13,37)(14,41)(16,46,47,43)(18,20,19,50)(24,52)(25,53)(26,55)(27,54)(30,56,57,63)(32,34,33,59)(35,39)(40,61)(45,60,58,62)(48,51,49,64);(1,5,8,27)(2,11,12,39)(3,16,17,47)(4,21,23,54)(6,26,7,25)(9,30,31,57)(10,35,36,38)(13,24,14,40)(15,43,44,46)(18,49,19,48)(20,51,50,64)(22,55,28,53)(29,56,42,63)(32,45,33,58)(34,60,59,62)(37,61,41,52)Polycyclic presentation:
2^9|a^2=f,b^2=efgh,c^2=f,d^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,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379878);