Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| 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 | |
661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,18)(11,19,22,33)(12,24,28,16)(13,29,30,17)(15,23,32,21)(20,37,39,34)(25,40,38,31)(26,35,27,36);(9,11,14,22)(10,15,18,32)(12,25)(13,23,30,21)(16,34)(17,33,29,19)(20,26)(24,37)(27,39)(28,38)(31,35)(36,40);(1,3,5,4)(2,6,7,8)(9,12,14,28)(10,16,18,24)(11,20,22,39)(13,26,30,27)(15,31,32,40)(17,35,29,36)(19,34,33,37)(21,25,23,38);(1,4,5,3)(2,6,7,8)(12,26)(13,30)(16,35)(17,29)(19,33)(20,25)(21,23)(24,36)(27,28)(31,34)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380813);