Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2075 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,17,12,19,27,18)(7,20,24,35,25,34,36,30)(8,16,26,15,10,13,9,28)(14,21,31,29,33,22,32,23);(1,2)(3,4)(6,13,19,16)(7,21,25,22)(8,26)(9,10)(11,27)(15,18,28,17)(20,35)(23,36,29,24)(30,34)(31,32);(2,4)(5,7,8,22,11,24,26,23,12,25,10,21,27,36,9,29)(6,14,15,30,18,32,16,34,19,33,28,35,17,31,13,20)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239860, #72240846, #72240857, … |
| maximal normal subgroup of | #72239860, #72240846, #72240857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56314);