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×#394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2542 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)(7,17,20,12)(8,21,23,32)(9,15)(10,16)(11,14)(13,24,30,26)(18,29)(19,31,25,28)(22,33)(27,35)(34,36);(5,7)(6,12)(8,22)(9,19)(10,25)(11,20)(13,29)(14,17)(15,28)(16,31)(18,24)(21,27)(23,35)(26,34)(30,36)(32,33);(2,4)(5,8)(6,13,14,30)(7,18)(9,24)(10,26)(11,23)(12,27,17,33)(15,21,16,32)(19,35)(20,34)(22,25)(28,36,31,29);(8,23)(13,30)(18,34)(21,32)(22,35)(24,26)(27,33)(29,36)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252199, #72252201, #72252209, … |
| maximal normal subgroup of | #72252199, #72252201, #72252209, … |
| maximal quotient of | #72314608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265124);