Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #205569 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,8)(6,7)(9,10,14,18,15,11,13,17)(12,20,23,19,21,24,22,16);(1,3,4,2,6,8,5,7)(9,11,15,10)(12,19,21,16)(13,17,14,18)(20,22,24,23);(9,12,13,22,15,21,14,23)(10,16,18,20,11,19,17,24);(1,4,6,5)(2,3,7,8)(12,21)(16,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,c^2=f,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286402, #72286414, #72286423, … |
| maximal normal subgroup of | #72286402, #72286414, #72286423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112929);