Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| 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 | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4334 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)(5,6,10,16,11,15,9,14)(7,17,19,28,21,12,20,27)(8,13,24,32,25,31,23,30)(18,33,22,36,34,26,35,29);(5,7)(6,12)(8,22)(9,19)(10,20)(11,21)(13,29)(14,27)(15,17)(16,28)(18,24)(23,34)(25,35)(26,32)(30,33)(31,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,30)(15,31)(16,32)(17,33)(19,22)(20,35)(21,34)(27,29)(28,36);(5,9,11,10)(6,14,15,16)(7,19,21,20)(8,23,25,24)(12,27,17,28)(13,30,31,32)(18,22,34,35)(26,29,33,36)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237510, #72237517, #72239618, … |
| maximal normal subgroup of | #72237510, #72237517, #72239618, … |
| maximal quotient of | #72249693, C2×#357567, #72316671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264763);