Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,7,11,15)(8,14,17,20,16,12,18,19);(5,7,9,15,10,6,11,13)(8,14,18,19,16,12,17,20);(5,8)(6,12)(7,14)(9,18)(10,16)(11,17)(13,20)(15,19);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^b=fi,e^c=ei,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72214896, C2×#392662, #72238734, … |
| maximal normal subgroup of | #72214896, C2×#392662, #72238734, … |
| maximal quotient of | #72210458, #72213768, #72214049, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299858);