Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2058 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,17,20,32)(8,23)(9,13)(10,27)(11,16)(12,19,28,18)(14,24)(15,25)(21,30,35,36)(22,29,34,33)(26,31);(5,7,8,19,10,22,9,21,11,20,24,18,25,34,26,35)(6,12,13,17,15,30,23,33,16,28,31,32,27,36,14,29);(1,2,3,4)(7,18,20,19)(8,9)(10,25)(12,17,28,32)(13,23)(14,31)(15,27)(21,34,35,22)(24,26)(29,36,33,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249891, #72249898, #72249927, … |
| maximal normal subgroup of | #72249891, #72249898, #72249927, … |
| maximal quotient of | #72264829 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56432);