Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393929 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(1,2)(3,4)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(2,4)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23);(6,8)Polycyclic presentation:
2^9|a^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240108, #72249909, #72251428, … |
| maximal normal subgroup of | #72240108, #72249909, #72251428, … |
| maximal quotient of | #72240127, #72249986, #72255822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301127);