Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,20,12)(9,14)(10,13)(15,24,31,35)(16,25)(17,26)(19,27,33,34)(21,30,22,28)(23,29)(32,36);(1,2,3,4)(5,7)(6,11)(8,19,20,33)(9,17)(10,16)(12,27,18,34)(13,25)(14,26)(15,21,31,22)(23,32)(24,28,35,30)(29,36);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,18)(14,30)(16,31)(17,33)(19,32)(21,23)(25,35)(26,34)(27,36)(28,29);(7,16)(11,25)(15,31)(17,32)(19,33)(24,35)(26,36)(27,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239315, #72239343, #72239427, … |
| maximal normal subgroup of | #72239315, #72239343, #72239427, … |
| maximal quotient of | #72239438, #72257367, #72257584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264952);