Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140598 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1961 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
736 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)(7,17,20,33)(8,14)(9,13)(10,25)(11,16)(12,18,28,21)(15,23)(19,32,36,27)(22,30)(24,31)(26,35,29,34);(1,2,3,4)(5,7)(6,12,16,28)(8,18)(9,19)(10,21)(11,20)(13,29,31,26)(14,17,25,33)(15,27,30,32)(22,34)(23,35)(24,36);(6,13)(8,10)(9,23)(12,26)(14,15)(16,31)(17,32)(18,21)(19,35)(22,24)(25,30)(27,33)(28,29)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,g^b=gi,d^c=di,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249862, #72249873, #72249902, … |
| maximal normal subgroup of | #72249862, #72249873, #72249902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55270);