Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,9,14,17)(8,11)(10,12)(13,16,19,20)(15,18);(5,7,9,12,10,14,17,6)(8,13,16,18,15,19,20,11);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20);(8,15)(11,18)(13,19)(16,20)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249448, #72249457, #72249466, … |
| maximal normal subgroup of | #72249448, #72249457, #72249466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402350);