Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,14,17,22)(15,24,25,20)(16,19,23,18)(21,26,28,27);(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,21)(17,25)(18,26)(19,27)(22,24)(23,28);(5,8,9,10)(6,7,11,12)(13,16,19,14,17,23,18,22)(15,20,26,21,25,24,27,28);(6,11)(7,12)(13,15,17,25)(14,21,22,28)(16,20,23,24)(18,27,19,26)Polycyclic presentation:
2^9|a^2=h,c^2=egi,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293093);