Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
939 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)(7,17,19,12)(8,13)(9,16)(10,14)(11,15)(18,26,34,33)(20,28,25,27)(21,35,36,29)(22,31)(23,30)(24,32);(1,2)(3,4)(5,7)(6,12)(8,21)(9,25)(10,20)(11,19)(13,29)(14,27)(15,17)(16,28)(18,23)(22,36)(24,34)(26,32)(30,33)(31,35);(2,4)(5,8)(6,13)(7,18,19,34)(9,23)(10,24)(11,22)(12,26,17,33)(14,32)(15,31)(16,30)(20,21,25,36)(27,35,28,29);(6,14,15,16)(8,22)(12,27,17,28)(13,30,31,32)(18,34)(21,36)(23,24)(26,29,33,35)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280214, #72280622, #72280664, … |
| maximal normal subgroup of | #72280214, #72280622, #72280664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305971);