Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,15)(11,16,12,14)(19,27)(20,21)(23,31)(24,25)(28,32)(29,30);(9,11,13,12)(10,14,15,16)(17,18,22,26)(19,28,30,31)(20,24,21,25)(23,27,32,29);(2,4)(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,29)(21,27)(22,30)(24,31)(25,28)(26,32);(6,8)(17,20,22,21)(18,24,26,25)(19,29,30,27)(23,31,32,28)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264627, #72264659, #72264675, … |
| maximal normal subgroup of | #72264627, #72264659, #72264675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302061);