Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| 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 | |
1354 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)(15,21,23,28)(16,24,26,19)(18,25,27,22);(9,10,12,11)(13,15)(14,18)(16,25)(17,23)(19,28)(20,27)(21,24)(22,26);(2,4)(5,6,8,7)(9,11,12,10);(5,7,8,6)(15,22)(16,26)(18,21)(19,24)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,d^2=ghi,e^2=i,b^a=be,c^a=cf,e^a=ei,d^b=dei,e^b=ei,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264359);