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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3456 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)(11,20)(12,17)(13,18)(14,19)(15,21)(16,23)(22,24);(5,6)(7,8)(9,11,12,21,14,22,13,23)(10,15,18,20,19,16,17,24);(2,4)(6,8)(9,11,12,21,14,22,13,23)(10,16,18,24,19,15,17,20);(6,8)(9,12,14,13)(10,17,19,18)(11,21,22,23)(15,24,16,20);(11,22)(15,16)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,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 | #72308271, #72308332, #72312918, … |
| maximal normal subgroup of | #72308271, #72308332, #72312918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346165);