Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6413 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)(5,6)(7,8)(10,11)(12,16)(17,18)(19,23)(20,21)(22,24);(6,7)(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,22,24,23);(6,7)(10,11)(14,15)(18,21)(22,23);(1,2,3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24);(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=h,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219257, #72305958, #72305960, … |
| maximal normal subgroup of | #72219257, #72219265, #72305958, … |
| maximal quotient of | #72328374, #72329899, #72330265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4364661);