Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,4)(3,10,11,16)(5,9,6,15)(8,12,14,13)(17,18)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3,6,13,7,11,5,12)(2,8,9,10,4,14,15,16)(18,22)(23,25)(24,29)(30,31);(3,11)(8,14)(10,16)(12,13)(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(1,4,7,2)(3,8,11,14)(5,9,6,15)(10,12,16,13)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(1,5,7,6)(2,9,4,15)(3,12,11,13)(8,10,14,16)(19,26)(24,29)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339707, #72339713, #72339791, … |
| maximal normal subgroup of | #72339707, #72339713, #72339791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264208);