Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| 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 | |
510 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)(5,6,10,16)(7,18)(8,21)(9,24,11,27)(12,30,25,17)(13,23)(14,28)(15,29,34,26)(19,33)(20,31)(22,32)(35,36);(2,4)(5,7,11,8,12,19,29,22,10,20,9,23,25,28,26,36)(6,13,15,31,17,32,24,33,16,21,34,18,30,35,27,14);(5,8,9,19,12,22,26,20,10,23,11,28,25,36,29,7)(6,14,15,21,17,31,24,35,16,33,34,13,30,18,27,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239807, #72239813, #72239831, … |
| maximal normal subgroup of | #72239807, #72239813, #72239831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54305);