Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2085 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
438 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)(5,6,11,18,13,19,12,17)(7,20,23,30,14,33,34,25)(8,24,27,35,28,22,26,21)(9,29,31,36,32,16,10,15);(5,7,12,34,13,14,11,23)(6,8,17,26,19,28,18,27)(9,24,10,21,32,22,31,35)(15,25,16,33,36,30,29,20);(5,8)(6,14,19,7)(9,30)(10,33)(11,26)(12,27)(13,28)(15,24,36,22)(16,35,29,21)(17,34,18,23)(20,31)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,h^b=hi,d^c=dhi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72239852, #72240850, … |
| maximal normal subgroup of | #72239851, #72239852, #72240850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56542);