Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(17,19);(2,4)(5,7,8,11)(6,9,10,12)(13,14)(15,19)(16,18)(17,20);(13,15)(14,17,18,19)(16,20);(6,10)(9,12)Polycyclic presentation:
2^9|b^2=h,c^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72235078, #72235081, #72260718, … |
| maximal normal subgroup of | #72235078, #72235081, #72260718, … |
| maximal quotient of | #72260795, #72317699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313468);