Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(14,20,23,24)(15,25)(16,27)(21,26,22,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,14,18,23)(15,26,25,28)(16,22,27,21)(17,24,19,20);(2,4)(6,10)(9,12)(13,15,17,16,18,25,19,27)(14,21,20,28,23,22,24,26);(13,16,17,25,18,27,19,15)(14,22,24,28,23,21,20,26)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72317237, #72317284, #72317291 |
| maximal normal subgroup of | #72317237, #72317284, #72317291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266534);