Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,8)(9,10,11,12)(13,14)(15,23)(16,19)(17,22)(18,21)(20,26)(24,27)(25,28);(1,2,4,3)(14,19)(15,24)(17,18)(20,23)(25,26)(27,28);(6,8)(13,15,16,25)(14,20,19,27)(17,24,18,26)(21,28,22,23);(1,3,4,2)(14,19)(20,27)(21,22)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72308259, #72308280, #72308297 |
| maximal normal subgroup of | #72308259, #72308280, #72308297 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267340);