Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,28)(22,32)(24,35)(26,34)(27,33)(29,31)(30,37)(36,38)(39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16,20,31)(12,22)(13,23)(14,29)(15,24)(17,32,33,30)(18,21,19,35)(25,36,34,40)(26,37)(27,38)(28,39);(9,12)(10,17,20,33)(11,22)(13,27)(14,28)(15,26)(16,30,31,32)(18,34,19,25)(21,36,35,40)(23,38)(24,37)(29,39);(2,6)(5,8)(12,26)(17,33)(22,37)(25,34)(27,28)(30,32)(36,40)(38,39)Polycyclic presentation:
2^9|b^2=gh,c^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317836, #72318245, #72318340, … |
| maximal normal subgroup of | #72317836, #72318245, #72318340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318027);