Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #61234 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,17,14,11,18,16)(12,15,22,24,23,19,21,20);(2,7)(6,8)(9,11)(10,14)(12,20)(13,16)(15,22)(17,18)(19,21)(23,24);(1,2)(3,6)(4,8)(5,7);(1,3,5,4)(2,6,7,8)(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309560, #72309568, #72309578, … |
| maximal normal subgroup of | #72309560, #72309568, #72309578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128240);