Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1046 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)(7,18,19,33)(8,14,24,34)(9,15,26,13)(10,28)(11,17)(12,20)(16,25)(21,31)(22,29,35,30)(23,32)(27,36);(2,4)(5,7,10,22,11,19,25,35)(6,12,16,31,17,32,28,36)(8,23,26,21,24,20,9,27)(13,30,14,33,15,29,34,18);(6,13)(7,19)(8,24)(9,26)(12,29)(14,16)(15,17)(18,31)(22,35)(28,34)(30,32)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72311838, #72311842, #72311867, … |
| maximal normal subgroup of | #72311838, #72311842, #72311867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53909);