Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1590 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)(5,6,9,16,11,14,10,15)(7,17,20,31,21,12,19,27)(8,22,24,13,23,30,25,29)(18,32,35,26,33,36,34,28);(5,7,10,20,11,21,9,19)(6,12,15,27,14,17,16,31)(8,18,25,35,23,33,24,34)(13,28,22,32,29,26,30,36);(1,2,3,4)(5,8,9,24,11,23,10,25)(6,13,15,22,14,29,16,30)(7,18,19,34,21,33,20,35)(12,26,27,36,17,28,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314120, #72314124, #72314126, … |
| maximal normal subgroup of | #72314120, #72314124, #72314126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31095);