Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,9,14,11,16,10,15)(7,17,20,12,19,29,21,28)(8,22,25,31,26,13,24,32)(18,30,35,27,34,33,23,36);(5,7)(6,12,16,28)(8,23,26,35)(9,20)(10,21)(11,19)(13,30)(14,29,15,17)(18,24,34,25)(22,33)(27,32)(31,36);(5,8,9,24,11,26,10,25)(6,13,14,31,16,22,15,32)(7,18,20,23,19,34,21,35)(12,27,29,30,28,36,17,33);(1,3,4,2)(6,14,16,15)(7,19)(12,17,28,29)(13,31,22,32)(18,34)(20,21)(23,35)(27,33,36,30)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=fgh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72268618 |
| maximal normal subgroup of | #72268618 |
| maximal quotient of | #72268632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286913);