Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,19,23,21);(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24);(2,4)(5,6,7,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,g^2=i,b^a=bd,c^a=ce,g^a=gi,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72219667, #72226701, #72226702, … |
| maximal normal subgroup of | #72219667, #72226701, #72226702, … |
| maximal quotient of | #72219659, #72226697, #72226698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30623);