Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,32)(25,30)(26,31);(9,10)(11,14)(12,13)(15,16)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32);(3,8)(5,7)(10,13)(14,16)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,30,31,32);(9,11,12,15)(10,14,13,16)(19,27)(23,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72257358, #72257572, #72257681 |
| maximal normal subgroup of | #72257358, #72257572, #72257681 |
| maximal quotient of | #72308461 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050522);