Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,13,11,14,9,12)(7,15,16,18,17,20,19,8);(1,2)(3,4)(5,7)(6,8)(9,16)(10,19)(11,17)(12,15)(13,20)(14,18);(2,4)(5,8)(6,7)(9,15)(10,20)(11,18)(12,16)(13,19)(14,17);(2,4);(5,9,11,10)(6,12,14,13)(7,16,17,19)(8,15,18,20)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312325, D4×#3018, #72312352, … |
| maximal normal subgroup of | #72312325, D4×#3018, #72312352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345158);