Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #3186 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,4,5,10)(3,11,14,12,15,16,13,8)(18,22,24,27)(19,25,26,28);(1,3,7,15)(2,8,4,12)(5,14,6,13)(9,16,10,11)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,29)(27,31);(1,5,7,6)(2,9,4,10)(3,13,15,14)(8,11,12,16)(17,20,21,29)(18,22,24,27)(19,25,26,28)(23,31,32,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102665);