Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,12,14,11,7,10,8)(9,16,17,13,18,20,19,15);(1,3,4,2)(5,7,11,6)(8,10,14,12)(9,15,18,13)(16,19,20,17);(5,8,10,6,11,14,12,7)(9,16,19,13,18,20,17,15);(5,9)(6,13)(7,15)(8,16)(10,19)(11,18)(12,17)(14,20)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227326, #72312195, C4×#24472 |
| maximal normal subgroup of | #72227326, #72312195, C4×#24472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277352);