Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 | #34737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3348 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)(9,10,14,19)(11,20,21,24)(12,18,13,17)(15,23,16,22);(5,6)(7,8)(9,11,12,22,14,21,13,23)(10,15,17,20,19,16,18,24);(2,4)(6,8)(9,11,12,22,14,21,13,23)(10,16,17,24,19,15,18,20);(6,8);(11,21)(15,16)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308312, #72308332, #72312904, … |
| maximal normal subgroup of | #72308312, #72308332, #72312904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346164);