Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4814 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)(9,10,15,19,14,11,13,17)(12,16,22,24,20,18,21,23);(5,6)(7,8)(9,11,14,10)(12,18,20,16)(13,17,15,19)(21,23,22,24);(2,4)(6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(12,20)(16,18)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305327, #72305338, #72305344, … |
| maximal normal subgroup of | #72305327, #72305338, #72305344, … |
| maximal quotient of | #72306160, #72306161, #72306162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344969);