Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4470 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)(5,6,10,15,11,16,9,14)(7,17,20,12,21,28,19,27)(8,22,23,31,25,13,24,30)(18,33,34,26,35,32,36,29);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,29)(14,17)(15,28)(16,27)(22,26)(23,34)(24,36)(25,35)(30,33)(31,32);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,30)(15,31)(16,22)(17,32)(19,34)(20,36)(21,35)(27,29)(28,33);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,27,28)(13,30,22,31)(18,34,35,36)(26,32,29,33)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237696, #72237702, #72237712, … |
| maximal normal subgroup of | #72237696, #72237702, #72237712, … |
| maximal quotient of | #72249805, #72281054, #72303314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264674);