Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,12,17,10,16)(7,18,21,13,22,28,26,29)(8,23,9,14,25,30,24,31)(19,33,20,32,34,27,35,36);(1,2,3,4)(5,7)(6,13)(8,19)(9,20)(10,26)(11,21)(12,22)(14,27)(15,28)(16,18)(17,29)(23,32)(24,35)(25,34)(30,36)(31,33);(5,8,11,9,12,25,10,24)(6,14,16,23,17,31,15,30)(7,19,21,20,22,34,26,35)(13,27,18,32,29,33,28,36);(5,9,11,25,12,24,10,8)(6,14,15,30,17,31,16,23)(7,20,21,34,22,35,26,19)(13,27,28,36,29,33,18,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224865, #72224903, #72224916, … |
| maximal normal subgroup of | #72224865, #72224903, #72224916, … |
| maximal quotient of | #72224856, #72239568, #72279972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261838);