Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,13,15)(11,14,20,22)(12,21,19,17)(16,24,23,18);(1,2)(3,6)(4,5)(7,8);(10,14)(11,18)(12,19)(15,22)(16,17)(20,24)(21,23);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,15)(14,22)(16,23)(17,21)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285760, #72287679, #72287708, … |
| maximal normal subgroup of | #72285760, #72287679, #72287708, … |
| maximal quotient of | #72285952, #72286953, #72343620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531271);