Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,13)(8,12)(9,11)(10,14)(15,20)(16,18)(17,19);(5,7)(6,10)(8,16,17,14)(9,15)(11,18)(12,20,19,13);(1,3,4,2)(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,20)(17,19);(6,11)(10,18)(12,19)(13,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,b^a=be,d^a=di,g^a=gh,e^b=eh,h^b=hi,d^c=di,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249433, #72249448, #72249457, … |
| maximal normal subgroup of | #72249433, #72249448, #72249457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400225);