Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,11)(7,15,17,33)(8,19)(9,20)(12,26,28,36)(13,30)(14,25)(16,32,18,21)(22,31)(23,27,29,34);(1,2,3,4)(5,6,8,11,10,14,19,25)(7,12,16,23,17,28,18,29)(9,13,20,24,22,30,31,35)(15,27,21,26,33,34,32,36);(5,7)(6,12)(8,16)(9,21)(10,17)(11,23)(13,26)(14,28)(15,31)(18,19)(20,33)(22,32)(24,34)(25,29)(27,35)(30,36);(7,16,17,18)(9,22)(12,23,28,29)(13,30)(15,32,33,21)(20,31)(24,35)(26,27,36,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=dh,f^c=fhi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226790, #72226795, #72226796, … |
| maximal normal subgroup of | #72226790, #72226795, #72226796, … |
| maximal quotient of | #72285417, #72285438, #72309432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403043);