Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
728 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)(7,15,17,32)(8,20)(9,24)(10,13)(11,18,16,34)(12,22)(14,26)(19,28,25,33)(21,30,36,27)(23,31)(29,35);(1,2,3,4)(5,6)(7,16,17,11)(8,12)(9,14)(10,13)(15,18,32,34)(19,30,25,27)(20,22)(21,33,36,28)(23,29)(24,26)(31,35);(5,7)(6,11)(8,21)(9,25)(10,17)(12,28)(13,16)(14,30)(15,31)(18,35)(19,26)(20,32)(22,34)(23,36)(24,27)(29,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ehi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224865, #72226807, #72226813, … |
| maximal normal subgroup of | #72224865, #72226807, #72226813, … |
| maximal quotient of | #72224861, #72237295, #72279513 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55260);