Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #15123 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1844 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)(7,10)(8,9)(11,12)(21,22)(23,27)(24,26)(25,28);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(22,26)(27,28);(2,4)(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,23,24,25)(22,27,26,28);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(22,28)(23,25)(26,27)Polycyclic presentation:
2^9|c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239711, #72239715, #72239717, … |
| maximal normal subgroup of | #72239711, #72239715, #72239717, … |
| maximal quotient of | #72257222, #72257246, #72314205 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376460);