Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,13,9,10,17,11)(7,14,15,18,16,12,19,20);(1,2)(3,4)(6,10)(8,17)(12,14)(15,19);(2,4)(6,11)(8,17)(10,13)(12,18)(14,20)(15,19);(5,7)(6,12,10,14)(8,15)(9,16)(11,18,13,20)(17,19)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260544, #72317836, #72318294, … |
| maximal normal subgroup of | #72260544, #72317836, #72318294, … |
| maximal quotient of | #72260620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303303);