Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,24,26,25,27,23,28);(6,9)(10,11)(13,15)(14,17)(16,20)(18,19);(2,4)(15,20)(17,19)(22,26)(23,24)(27,28);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(22,27)(26,28)Polycyclic presentation:
2^9|a^2=h,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264206, #72264239, #72315981, … |
| maximal normal subgroup of | #72264206, #72264239, #72315981, … |
| maximal quotient of | #72265179 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270591);