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×#2213 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
721 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)(7,16,18,33)(8,12)(9,20)(13,27,29,36)(14,31)(15,25)(17,32,19,21)(22,26)(23,28,30,34);(1,2,3,4)(5,6,12,25,10,15,8,11)(7,13,19,30,18,29,17,23)(9,14,26,35,22,31,20,24)(16,28,32,36,33,34,21,27);(5,7)(6,13)(8,17)(9,21)(10,18)(11,23)(12,19)(14,27)(15,29)(16,26)(20,33)(22,32)(24,34)(25,30)(28,35)(31,36);(7,17,18,19)(9,22)(13,23,29,30)(14,31)(16,32,33,21)(20,26)(24,35)(27,28,36,34)Polycyclic presentation:
2^9|a^2=g,b^2=ei,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 | #72226791, #72226795, #72226796, … |
| maximal normal subgroup of | #72226791, #72226795, #72226796, … |
| maximal quotient of | #72285422, #72285451, #72309584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403044);