Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2023 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
710 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)(7,17,19,33)(8,14)(9,26)(10,28)(11,16)(12,20,30,35)(13,24)(15,27)(18,22,32,29)(21,34,36,31)(23,25);(1,2,3,4)(5,6)(7,18,19,32)(8,23)(9,13)(10,28)(11,16)(12,21,30,36)(14,25)(15,27)(17,29,33,22)(20,31,35,34)(24,26);(5,7)(6,12)(8,21)(9,20)(10,29)(11,19)(13,32)(14,33)(15,34)(16,30)(17,23)(18,26)(22,27)(24,35)(25,36)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249898, #72249912, #72249928, … |
| maximal normal subgroup of | #72249898, #72249912, #72249928, … |
| maximal quotient of | #72265021, #72279642 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55320);