Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,25,30,32)(20,31,29,24)(21,27,22,26);(5,6)(7,8)(17,19,23,30)(18,24,28,31)(20,21,29,22)(25,27,32,26);(2,4)(6,8)(17,20,22,19,23,29,21,30)(18,25,27,31,28,32,26,24);(9,11,12,16)(10,13,14,15)(17,21,23,22)(18,26,28,27)(19,29,30,20)(24,25,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264155, #72264172, #72264316, … |
| maximal normal subgroup of | #72264155, #72264172, #72264316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322076);