Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
16927 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)(5,6)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18);(2,4)(9,10)(11,15)(12,21)(13,16)(14,17)(18,19)(20,24)(22,23);(6,8);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,f^b=fi,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72313597, #72313623, #72313664, … |
| maximal normal subgroup of | #72313597, #72313623, #72313664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536054);