Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482362 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,19,20,24)(12,18,13,17)(14,16)(15,21)(22,23);(1,2)(3,6)(4,8)(5,7)(9,11,14,20)(10,15,16,23)(12,21,13,22)(17,19,18,24);(1,3,5,4)(2,6,7,8)(10,16)(11,20)(17,18)(21,22);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,16,18)(11,21,20,22)(15,19,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72281359, #72281372, #72281603, … |
| maximal normal subgroup of | #72281359, #72281372, #72281603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389833);