Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535641 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(17,18,21,23)(19,25,26,32)(20,28,29,24)(22,30,31,27);(9,10)(11,14)(12,13)(15,16)(17,19,20,27,21,26,29,30)(18,22,24,25,23,31,28,32);(2,5)(6,8)(10,13)(14,16)(18,23)(22,31)(24,28)(25,32);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,26)(22,31)(25,32)(27,30);(11,15)(14,16)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256896, #72256898, #72256903, … |
| maximal normal subgroup of | #72256896, #72256898, #72256903, … |
| maximal quotient of | #72265600, #72265624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6453595);