Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,12,11)(13,14,19,24)(15,21,26,28)(16,27,25,20)(17,23,18,22);(5,6)(7,8)(13,15,19,26)(14,20,24,27)(16,17,25,18)(21,23,28,22);(2,4)(6,8)(13,16,18,15,19,25,17,26)(14,21,23,27,24,28,22,20);(9,11,12,10)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264369, #72264372, #72264377, … |
| maximal normal subgroup of | #72264369, #72264372, #72264377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266316);