Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,17,13,18,12,21)(7,19,22,29,15,9,16,10)(8,23,27,20,24,35,28,32)(14,30,33,36,31,25,34,26);(1,2,3,4)(5,7,19,21,12,22,10,6,13,15,9,17,11,16,29,18)(8,20,30,34,28,35,26,14,24,32,25,33,27,23,36,31);(5,8,9,25,12,28,29,36,13,24,19,30,11,27,10,26)(6,14,7,20,17,33,22,35,18,31,15,32,21,34,16,23);(8,24)(14,31)(20,32)(23,35)(25,30)(26,36)(27,28)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,c^2=fh,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=ei,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239801, #72239813, #72239838, … |
| maximal normal subgroup of | #72239801, #72239813, #72239838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402430);