Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1244 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)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(10,13)(14,16)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(2,4)(9,11,12,15)(10,14,13,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(5,6,7,8)(18,23)(22,30)(24,32)(29,31)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264623, #72264657, #72264782, … |
| maximal normal subgroup of | #72264623, #72264657, #72264782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319780);