Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3367 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)(10,11)(13,14)(16,19)(17,18)(20,24)(22,23);(5,6)(7,8)(9,10,13,11,15,17,14,18)(12,20,22,24,21,16,23,19);(2,4)(6,8)(9,11,13,17,15,18,14,10)(12,19,22,20,21,24,23,16);(6,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,21)(17,20)(18,24);(10,17)(11,18)(12,21)(22,23)Polycyclic presentation:
2^9|b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315645, #72315668, #72315838, … |
| maximal normal subgroup of | #72315645, #72315668, #72315838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346199);