Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,8,7)(10,15)(11,19)(13,26)(16,30)(17,33)(18,27)(21,36)(22,31)(23,39)(24,28)(29,35)(34,37);(1,2)(3,4)(5,7,8,6)(9,10,14,18)(11,20,22,38)(12,17,25,35)(13,27,26,15)(16,31,30,19)(21,37,39,34)(23,40,36,32)(24,29,28,33);(2,4)(6,7)(9,11,13,19,14,22,26,31)(10,16,15,20,18,30,27,38)(12,23,24,39,25,36,28,21)(17,34,33,40,35,37,29,32);(9,12,14,25)(10,17,18,35)(11,21,22,39)(13,28,26,24)(15,29,27,33)(16,32,30,40)(19,36,31,23)(20,37,38,34)Polycyclic presentation:
2^9|a^2=efi,b^2=ef,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364098);