Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2110 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)(18,22)(19,25)(21,26)(23,30)(27,31)(28,32);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,31)(26,29)(27,28)(30,32);(11,16)(13,15)(17,19,26,32,20,28,21,25)(18,23,29,30,22,27,24,31);(2,5)(6,8)(10,14)(13,15)(17,18)(19,27)(20,22)(21,24)(23,28)(25,30)(26,29)(31,32);(1,3,4,7)(2,6,5,8)(18,22)(19,28)(24,29)(25,32)Polycyclic presentation:
2^9|c^2=gi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251206, #72251228, #72256882, … |
| maximal normal subgroup of | #72251206, #72251228, #72256882, … |
| maximal quotient of | #72308835, #72308872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6704825);