Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| 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 | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,23)(12,29)(13,32)(14,21)(15,20)(16,22)(17,26)(18,30)(19,34)(24,35)(25,40)(27,38)(28,37)(31,36)(33,39);(1,3,5,8)(2,6,4,7)(9,11,14,27,16,28,15,26)(10,17,21,23,22,38,20,37)(12,24,30,13,31,33,25,34)(18,39,36,19,40,35,29,32);(9,12,15,30,16,31,14,25)(10,18,21,29,22,40,20,36)(11,24,27,34,28,33,26,13)(17,35,37,32,38,39,23,19);(1,4,5,2)(3,6,8,7)(9,13,14,33,16,34,15,24)(10,19,20,39,22,32,21,35)(11,25,26,31,28,30,27,12)(17,36,23,40,38,29,37,18)Polycyclic presentation:
2^9|a^2=ef,b^2=eg,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 345962);