Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,12,14,27)(7,16)(8,20)(9,10)(13,26,29,28)(15,31,25,21)(17,32)(22,34)(23,24)(30,33,35,36);(5,6,9,15,11,14,10,25)(7,13,18,30,16,29,19,35)(8,21,23,27,24,31,20,12)(17,33,32,26,22,36,34,28);(1,3,4,2)(5,7,10,19,11,16,9,18)(6,13,25,35,14,29,15,30)(8,22,20,32,24,17,23,34)(12,26,31,33,27,28,21,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320900, #72320995, #72321026, … |
| maximal normal subgroup of | #72320900, #72320995, #72321026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54436);