Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137655 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| 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 | |
680 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)(5,6,9,16,10,15,25,23,12,17,26,14,11,8,21,24)(7,13,18,29,27,31,36,32,20,30,34,28,19,22,33,35);(1,2,3,4)(5,7)(6,13)(8,22)(9,18)(10,27)(11,19)(12,20)(14,28)(15,31)(16,29)(17,30)(21,33)(23,32)(24,35)(25,36)(26,34);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,26)(18,32)(19,31)(20,30)(21,23)(22,27)(24,25)(28,33)(29,36)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239809, #72239813, #72239819, … |
| maximal normal subgroup of | #72239809, #72239813, #72239819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55397);