Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1446 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,17,11)(8,12)(9,14)(10,13)(16,23,31,29)(18,30,32,24)(19,25,33,35)(20,27)(21,26)(22,28)(34,36);(5,7)(6,11)(8,19,20,33)(9,18)(10,17)(12,25,27,35)(13,15)(14,24)(16,21,31,34)(22,32)(23,26,29,36)(28,30);(1,2,3,4)(5,8)(6,12)(7,16,17,31)(9,21)(10,20)(11,23,15,29)(13,27)(14,26)(18,19,32,33)(22,34)(24,25,30,35)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,g^b=gi,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210904, #72239315, #72239326, … |
| maximal normal subgroup of | #72210904, #72239315, #72239326, … |
| maximal quotient of | #72210873, #72257362, #72257579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31112);