Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C42 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(13,14,16,22,17,19,25,26,18,20,24,27,15,21,23,28);(2,4)(5,6,7,9,8,10,11,12)(14,19,20,21)(16,24)(22,28,27,26)(23,25);(14,20)(19,21)(22,27)(26,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,f^a=fi,e^b=ei176 conjugacy classes — view the full character table.
| maximal subgroup of | #72285924, #72285938, #72285986 |
| maximal normal subgroup of | #72285924, #72285938, #72285986 |
| maximal quotient of | #72286531 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33746);