Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2011 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(18,23)(19,24)(20,21);(2,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,23)(22,24);(6,8)(10,13)(14,16)(17,18)(19,20)(21,23)(22,24);(9,11,12,15)(10,14,13,16)(17,19,22,23)(18,20,24,21)Polycyclic presentation:
2^9|d^2=gi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264374, #72264385, #72264507, … |
| maximal normal subgroup of | #72264374, #72264385, #72264507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311129);