Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464719 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,21)(9,12)(10,16)(11,15)(14,24)(17,27)(18,32)(19,29)(20,25)(22,30)(23,26)(28,34)(31,36)(33,35);(6,12)(7,17,20,19)(13,27,25,29)(14,26)(15,16)(18,31,33,34)(21,30)(28,32,36,35);(5,7)(6,13)(8,18,22,33)(9,20)(10,19)(11,17)(12,25)(14,28,26,36)(15,27)(16,29)(21,35,30,32)(23,34,24,31);(2,4)(5,8,11,24,9,22,10,23)(6,14,16,21,12,26,15,30)(7,18,17,31,20,33,19,34)(13,28,29,35,25,36,27,32)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285151, #72285452, #72285492, … |
| maximal normal subgroup of | #72285151, #72285452, #72285492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376969);