Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232181 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15,18,11)(8,12)(9,14)(10,13)(16,19,17,20);(1,2)(3,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,19)(13,15)(14,20);(2,4)(6,12)(7,16)(8,9)(11,15)(13,14)(17,18);(7,17)(8,9)(11,19)(12,14)(15,20)(16,18)Polycyclic presentation:
2^9|a^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284684, #72285522, #72285613, … |
| maximal normal subgroup of | #72284684, #72285522, #72285613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298413);