Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,13,22,27)(7,15,18,28,17,11,21,24)(8,12,20,26,19,25,32,35)(16,23,31,34,30,29,33,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,21)(10,17)(12,23)(13,15)(14,28)(18,22)(19,30)(20,33)(24,27)(25,29)(26,36)(31,32)(34,35);(5,8,9,20,10,19,22,32)(6,12,14,26,13,25,27,35)(7,16,21,33,17,30,18,31)(11,23,28,36,15,29,24,34);(6,13)(11,15)(12,25)(14,27)(23,29)(24,28)(26,35)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^b=fi,g^b=gi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72251311, #72251316, #72251323, … |
| maximal normal subgroup of | #72251311, #72251316, #72251323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261231);