Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,20,23,21,24,19,22);(9,10)(11,15)(12,13)(14,16)(18,22)(19,20)(23,24);(2,6)(7,8)(10,13)(14,15)(17,19,21,20)(18,23,24,22);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19,21,20)(18,23,24,22);(1,4,5,3)(2,8,6,7)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=g,e^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224777, #72224781, #72224785, … |
| maximal normal subgroup of | #72224777, #72224781, #72224785, … |
| maximal quotient of | #72224851, #72224853, #72224915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376926);