Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2177 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)(5,6,12,14)(7,15,18,19)(8,10,20,11)(9,16,13,17);(5,7)(6,13)(8,19)(9,14)(10,17)(11,16)(12,18)(15,20);(5,8,10,14,12,20,11,6)(7,9,17,15,18,13,16,19);(5,9,12,13)(6,7,14,18)(8,17,20,16)(10,15,11,19)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227844, #72312189, C4×#24472, … |
| maximal normal subgroup of | #72227844, #72312189, C4×#24472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267788);