Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,15,19,14,11,13,17)(12,18,22,23,20,16,21,24);(5,6)(7,8)(9,11,14,10)(12,16,20,18)(13,17,15,19)(21,24,22,23);(1,2,3,4)(6,8)(9,12,14,20)(10,16,11,18)(13,21,15,22)(17,23,19,24);(6,8)(10,11)(16,18)(17,19)(23,24);(12,20)(16,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312751, #72312881, #72312949 |
| maximal normal subgroup of | #72312751, #72312881, #72312949 |
| maximal quotient of | #72315780 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350199);