Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,11)(12,15,16,14)(17,18,20,23)(19,25,27,22)(21,24,29,31)(26,30,28,32);(2,5)(6,7)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,32)(28,29)(30,31);(9,12,13,16)(10,14,11,15)(18,23)(22,25)(24,31)(30,32);(10,11)(14,15)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,3,4,8)(2,6,5,7)(18,23)(19,27)(24,31)(26,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72313261, #72313513, #72313679, … |
| maximal normal subgroup of | #72313261, #72313513, #72313679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7452782);