Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5196 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)(11,19)(12,22)(13,17)(14,20)(15,18)(16,23)(21,24);(5,6)(7,8)(9,11,14,22,15,21,13,16)(10,12,17,19,18,23,20,24);(1,2,3,4)(6,8)(9,12,15,23)(10,16,18,22)(11,20,21,17)(13,24,14,19);(6,8)(11,21)(12,23)(16,22)(19,24);(9,13,15,14)(10,17,18,20)(11,22,21,16)(12,24,23,19)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281400, #72281501, #72281914, … |
| maximal normal subgroup of | #72281400, #72281501, #72281914, … |
| maximal quotient of | #72305807, #72316213 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350301);