Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #37439 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,17)(16,19)(18,20);(1,3,4,2)(6,10)(7,8)(11,12)(17,20);(5,7,9,8)(6,11,12,10)Polycyclic presentation:
2^9|c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309178, #72309181, #72309223, … |
| maximal normal subgroup of | #72309178, #72309181, #72309223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145677);