Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(10,14)(13,15)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(18,25)(20,21)(23,27)(24,26)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=f,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313243, #72313366, #72313564, … |
| maximal normal subgroup of | #72313243, #72313366, #72313564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6648962);