Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,18,13,19,12,32)(7,20,23,36,25,15,24,33)(8,26,22,35,28,14,27,34)(9,29,10,31,21,17,30,16);(2,4)(5,7,11,24,13,25,12,23)(6,14,18,35,19,26,32,34)(8,21,27,30,28,9,22,10)(15,31,36,17,20,16,33,29);(5,8)(6,15,19,20)(7,21)(9,25)(10,23)(11,27)(12,22)(13,28)(14,31,26,16)(17,34,29,35)(18,36,32,33)(24,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239850, #72239860, #72239861, … |
| maximal normal subgroup of | #72239850, #72239860, #72239861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54500);