Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | #23587 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,8)(5,10)(6,9)(7,11)(13,16)(14,15)(17,18,22,19,21,24,30,26)(20,23,29,25,28,31,32,27);(1,3,7,13)(2,4,11,14)(5,15,6,8)(9,16,10,12)(17,19,22,24,21,26,30,18)(20,27,29,23,28,25,32,31);(1,4,7,14)(2,3,11,13)(5,12,6,16)(8,10,15,9)(17,20)(18,23)(19,25)(21,28)(22,29)(24,31)(26,27)(30,32);(1,4)(2,8)(3,9)(5,16)(6,12)(7,14)(10,13)(11,15)(18,24)(19,26)(20,28)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,d^c=dhi,f^c=fh,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72215940 |
| maximal normal subgroup of | #72215940 |
| maximal quotient of | #72215902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134177);