Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235000 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,9)(5,19)(6,11)(7,12)(8,15)(10,22)(14,26)(16,27)(17,30)(18,28)(20,25)(21,29)(23,32)(24,31);(1,3)(2,8)(4,15)(5,20)(6,16)(7,17)(9,27)(10,23)(11,25)(12,26)(13,19)(14,18)(21,32)(22,31)(24,28)(29,30);(4,18)(5,21)(8,23)(9,28)(13,27)(14,15)(19,29)(20,32)(24,30)(25,26);(1,4,6,5)(2,9,11,19)(3,14,16,32)(7,18,22,21)(8,24,25,30)(10,29,12,28)(13,26,27,23)(15,31,20,17)Polycyclic presentation:
2^9|d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283074 |
| maximal normal subgroup of | #72283074 |
| maximal quotient of | #72284678 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142198);