Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2322 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15);(11,16)(13,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,31)(26,32);(2,5)(6,8)(10,14)(13,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(1,3,4,7)(2,6,5,8)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,31,32,24)Polycyclic presentation:
2^9|e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72281004, #72281013, #72281251, … |
| maximal normal subgroup of | #72281004, #72281013, #72281251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699645);