Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,10)(7,11,8,12)(13,14,15,17)(16,18,20,19);(1,2)(3,4)(6,10)(11,12)(14,17)(18,19);(2,4)(14,18)(16,20)(17,19);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(14,17)(18,19);(2,4)(5,8,9,7)(6,12,10,11)(14,17)(18,19)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72280241, #72280484, D4×#3109, … |
| maximal normal subgroup of | #72280241, #72280484, D4×#3109, … |
| maximal quotient of | #72304910, #72305532, #72305541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485444);