Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196248 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,12,8,18)(7,14,17,19,16,20,15,11);(5,7)(6,11,12,19)(8,15)(9,17)(10,16)(13,14,18,20);(6,12)(8,9)(11,19)(15,17);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210511, #72216077, #72216130, … |
| maximal normal subgroup of | #72210511, #72216077, #72216130, … |
| maximal quotient of | #72213768, #72213796, C2×#392696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299892);