Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,7)(9,10,12,11)(13,14,17,20)(15,21,24,27)(16,25,26,19)(18,22,28,23);(9,10,12,11)(13,15,17,24)(14,18,20,28)(16,22,26,23)(19,27,25,21);(2,4)(5,6,8,7)(9,11,12,10);(5,7,8,6)(15,22)(16,26)(18,27)(19,25)(21,28)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=gh,e^2=i,b^a=be,c^a=cf,e^a=ei,d^b=de,e^b=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264350);