Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2206 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,13,18)(11,15,14,22)(12,16,21,24)(17,19,23,20);(1,3,4,2)(6,7)(9,11,13,14)(10,15,18,22)(12,19,21,20)(16,23,24,17);(10,16)(11,14)(12,20)(15,17)(18,24)(19,21)(22,23);(5,7,8,6)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2177, #72226877, #72226892, … |
| maximal normal subgroup of | D4×#2177, #72226877, #72226892, … |
| maximal quotient of | #72225558, #72225559, #72226877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405502);