Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438013 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2410 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4565 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)(3,12,16,8)(4,9)(5,20)(6,10)(7,11)(13,15,24,21)(14,27,30,23)(17,18)(19,26)(22,25)(28,29,32,31);(1,2)(3,13)(4,17)(5,10)(6,20)(7,11)(8,15)(9,18)(12,21)(14,23)(16,24)(19,25)(22,26)(27,30)(28,31)(29,32);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,23)(10,24)(11,12)(13,20)(17,28)(18,29)(19,30)(22,31)(25,32)(26,27);(1,4)(2,9)(3,14)(5,18)(6,22)(7,19)(8,23)(10,25)(11,26)(12,27)(13,28)(15,29)(16,30)(17,20)(21,31)(24,32);(1,5,7,6)(2,10,11,20)(3,15,16,21)(4,18,19,22)(8,24,12,13)(9,25,26,17)(14,29,30,31)(23,32,27,28)Polycyclic presentation:
2^9|a^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,f^c=fi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72219682, #72219687, … |
| maximal normal subgroup of | #72215953, #72219682, #72219687, … |
| maximal quotient of | #72251071, #72251082, #72251308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345211);