Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | #35250 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10,15,11)(12,20,22,24)(13,17,14,19)(16,23,18,21);(1,2,4,3)(5,6)(7,8)(9,11,15,10)(12,18,22,16)(13,17,14,19)(20,21,24,23);(1,3,4,2)(6,8)(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,24)(19,20);(6,8);(9,13,15,14)(10,17,11,19)(12,21,22,23)(16,24,18,20)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315405, #72315432, #72315453 |
| maximal normal subgroup of | #72315405, #72315432, #72315453 |
| maximal quotient of | #72315685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345979);