Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2361 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(10,14,16,20)(11,17)(15,22,23,18)(19,24);(2,6)(5,8)(9,10,12,20,13,16,21,14)(11,18,19,15,17,22,24,23);(1,2,4,6)(3,8,7,5);(9,11,12,19,13,17,21,24)(10,15,14,18,16,23,20,22)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309482, #72309492, #72309504, … |
| maximal normal subgroup of | #72309482, #72309492, #72309504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 136434);