Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3406 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)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(1,2)(3,4)(5,6)(7,8);(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(2,4)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284945, #72285030, #72285048, … |
| maximal normal subgroup of | #72284945, #72285030, #72285048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408143);