Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| 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 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
691 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)(5,6)(7,17)(8,21,23,28)(9,25)(10,12)(11,16)(13,19,31,36)(14,18)(15,20)(22,34,24,33)(26,30,35,32)(27,29);(1,2,3,4)(5,7,18,27,11,20,9,10)(6,12,25,15,16,29,14,17)(8,19,24,35,23,36,22,26)(13,28,32,33,31,21,30,34);(5,8)(6,13)(7,19)(9,22)(10,26)(11,23)(12,28)(14,30)(15,33)(16,31)(17,34)(18,24)(20,36)(21,29)(25,32)(27,35);(7,20)(8,22,23,24)(10,27)(12,29)(13,30,31,32)(15,17)(19,35,36,26)(21,34,28,33)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^b=di,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 | #72224792, #72226795, #72226796, … |
| maximal normal subgroup of | #72224792, #72226795, #72226796, … |
| maximal quotient of | #72242475, #72242481, #72285410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403048);