Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
498 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)(6,13,16,17)(7,18)(8,22)(9,11)(10,23)(14,29,32,36)(15,33,24,28)(19,34,31,27)(20,26)(21,25);(2,4)(5,6,10,17,11,15,23,33,12,16,8,13,9,24,22,28)(7,19,20,36,18,32,35,34,21,31,26,29,25,14,30,27);(5,7)(6,14,16,32)(8,20)(9,25)(10,26)(11,18)(12,21)(13,27,17,34)(15,19,24,31)(22,35)(23,30)(28,36,33,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250498, #72250514, #72250519, … |
| maximal normal subgroup of | #72250498, #72250514, #72250519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54398);