Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2679 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3830 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)(6,7)(9,10,13,15)(11,16,12,14)(17,18,21,22)(19,23,24,20);(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,20,22,23);(5,6,8,7)(11,12)(14,16)(18,22)(20,23);(1,2)(3,4)(5,7,8,6)(9,12,13,11)(10,14,15,16)(17,20,21,23)(18,19,22,24);(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgh,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72323399, #72323415, #72323417, … |
| maximal normal subgroup of | #72323399, #72323415, #72323417, … |
| maximal quotient of | #72323642, C2×#4315167 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4222363);