Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1872 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)(7,14,15,10)(8,11)(9,12)(13,18)(16,20,17,19);(1,2)(3,4)(5,7)(6,10)(8,17)(9,15)(11,20)(12,14)(13,19)(16,18);(2,4)(10,14)(19,20);(6,11,12,13)(10,19,14,20)Polycyclic presentation:
2^9|a^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219662, #72219671, … |
| maximal normal subgroup of | #72216067, #72219662, #72219671, … |
| maximal quotient of | #72260596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327010);