Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(10,14)(15,23)(16,17)(18,24);(2,4)(9,10,12,14)(11,18,19,24)(13,16,21,17)(15,22,23,20);(5,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,23)(16,18)(17,24)(20,21);(10,16)(13,21)(14,17)(15,18)(20,22)(23,24);(11,19)(15,23)(18,24)(20,22)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312696, #72312881, #72312963, … |
| maximal normal subgroup of | #72312696, #72312881, #72312963, … |
| maximal quotient of | #72315786 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350051);