Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423425 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
704 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)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,28)(22,31)(23,38)(25,33)(27,34)(29,36)(30,35)(32,37)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,27)(13,25)(14,24)(15,22)(17,31)(18,34)(19,21)(20,33)(23,28)(26,32)(29,37)(30,39)(35,38)(36,40);(10,17,20,19)(11,22)(16,31,33,21)(18,26,36,35)(23,37)(24,25)(27,39)(32,40,38,34);(1,4,5,2)(3,6,8,7)(9,12,14,29,15,30,13,28)(10,18,17,26,20,36,19,35)(11,23,25,39,22,37,24,27)(16,32,21,34,33,38,31,40)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=gi,d^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356634);