Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7301204 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,23)(20,25)(21,24)(22,26)(27,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,32)(25,30)(26,31);(9,11)(10,13)(12,16)(14,15)(18,24)(19,27)(20,21)(23,30)(25,26)(31,32);(3,8)(5,7)(10,14)(11,16)(17,20,22,21)(18,25,26,24)(19,28,27,29)(23,30,31,32);(2,6)(5,7)(10,14)(13,15)(18,26)(19,27)(24,25)(28,29)Polycyclic presentation:
2^9|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72282000, #72282026, #72282263, … |
| maximal normal subgroup of | #72282000, #72282026, #72282263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7208638);