Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2338 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)(11,15)(12,14)(13,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,30)(28,31)(29,32);(9,11,12,16)(10,13,14,15)(18,23)(19,28)(21,22)(24,27)(29,30)(31,32);(2,4)(17,19)(18,24)(20,29)(21,28)(22,30)(23,31)(25,32)(26,27);(10,14)(13,15)(18,23)(19,29)(25,26)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264588, #72264604, #72264614, … |
| maximal normal subgroup of | #72264588, #72264604, #72264614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266947);