Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,23)(18,28)(20,31)(22,30)(24,29)(25,27)(26,34)(32,33)(35,36);(5,7,9,19,11,20,10,25)(6,12,14,17,16,27,15,31)(8,18,22,33,24,34,23,35)(13,28,29,36,30,26,21,32);(5,8,9,22,11,24,10,23)(6,13,14,29,16,30,15,21)(7,18,19,33,20,34,25,35)(12,26,17,32,27,28,31,36);(2,4)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307257, #72307337, #72307361, … |
| maximal normal subgroup of | #72307257, #72307337, #72307361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301944);