Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7513948 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2760 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11,14,7)(4,15,18,8)(5,9)(6,10)(12,24)(13,25,27,21)(16,20)(17,28,30,22)(19,23)(26,32)(29,31);(1,3)(2,7)(4,16)(5,13)(6,14)(8,20)(9,21)(10,11)(12,18)(15,24)(17,29)(19,27)(22,31)(23,25)(26,30)(28,32);(1,4)(2,8)(3,12,14,16)(5,17)(6,18)(7,20,11,24)(9,22)(10,15)(13,26,27,29)(19,30)(21,31,25,32)(23,28);(5,19)(7,11)(9,23)(13,27)(17,30)(20,24)(22,28)(26,29);(1,5,6,19)(2,9,10,23)(3,13,14,27)(4,17,18,30)(7,21,11,25)(8,22,15,28)(12,26,16,29)(20,31,24,32)Polycyclic presentation:
2^9|a^2=gh,c^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,g^b=gi,h^c=hi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284620, #72284711, #72284740, … |
| maximal normal subgroup of | #72284620, #72284711, #72284740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421145);