Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,17,14,18,12,16)(11,19,21,15,22,23,20,24);(1,2,3,4)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,19)(18,24);(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24);(6,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,23)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,g^b=gi,h^b=hi,f^c=fi,e^d=ei,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72283148, #72317682, #72317709, … |
| maximal normal subgroup of | #72283148, #72317682, #72317709, … |
| maximal quotient of | #72283236, #72319773, #72319898 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299850);