Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,15,18,11)(10,12)(13,20,16,17)(14,19);(1,2,4,3)(5,7)(6,9)(8,16,18,13)(10,14)(11,17,15,20)(12,19);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20);(1,3,4,2)(5,9)(6,7)(8,17)(10,19)(11,13)(12,14)(15,16)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225630, #72225634, … |
| maximal normal subgroup of | #72225620, #72225630, #72225634, … |
| maximal quotient of | #72251548, #72265804, #72265806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283376);