Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127185 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
919 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,20,23)(11,17,19,24,18,14,21,22);(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,24)(20,21)(22,23);(1,3,5,4)(2,7,8,6)(11,18)(14,17)(19,21)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ci,e^b=eh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231342, #72231376, #72231394, … |
| maximal normal subgroup of | #72231342, #72231376, #72231394, … |
| maximal quotient of | C2×#127209, #72261314, #72262212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34405);