Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2535 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(9,10)(11,12)(13,14,17,19,18,15,16,20);(2,3)(5,6)(7,8);(6,8)(13,15)(14,18)(16,20)(17,19);(1,3,4,2)(10,12)(13,16,18,17)(14,19,15,20)Polycyclic presentation:
2^9|a^2=ef,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72320123, #72320138, #72320325, … |
| maximal normal subgroup of | #72320123, #72320138, #72320325, … |
| maximal quotient of | #72321985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327170);