Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3755 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)(5,6,10,15,11,12,9,22)(7,13,16,26,18,25,23,28)(8,19,21,29,20,14,24,30)(17,27,33,36,32,34,31,35);(2,4)(6,12)(7,16)(9,10)(13,26)(14,19)(17,31)(18,23)(21,24)(25,28)(27,35)(32,33)(34,36);(5,7)(6,13)(8,17)(9,23)(10,16)(11,18)(12,25)(14,27)(15,26)(19,34)(20,32)(21,33)(22,28)(24,31)(29,35)(30,36);(2,4)(5,8)(6,14)(7,17)(9,21)(10,24)(11,20)(12,19)(13,27)(15,29)(16,31)(18,32)(22,30)(23,33)(25,34)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72311939, #72311981, #72312228, … |
| maximal normal subgroup of | #72311939, #72311981, #72312228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266740);