Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,20);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,17)(13,19)(16,20);(10,17)(18,19);(1,3,4,2)(6,11)(8,16)(10,18)(12,13)(15,20)(17,19)Polycyclic presentation:
2^9|d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72252458, #72252460, #72252462, … |
| maximal normal subgroup of | #72252458, #72252460, #72252462, … |
| maximal quotient of | #72260897, C2×#418314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325510);